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Limits Continuity and Differentiability question

2012 · Shift 1 · Q36
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  5. /2012 · Shift 1 · Q36

Limits Continuity and Differentiability question

2012 · Shift 1 · Q36

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMCQ+3 / −1
If lim⁡x→∞(x2+x+1x+1−ax−b)=4\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4x→∞lim​(x+1x2+x+1​−ax−b)=4, then
  1. A
    a = 1, b = 4
  2. B
    a = 1, b = −-− 4
  3. C
    a = 2, b = −-− 3
  4. D
    a = 2, b = 3
View written solutionFree

Correct answer: B

Step-by-step Solution:

  1. Start with the given limit expression: We are given the equation:

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opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opage-bre opagebreak opage-bre opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opage-break opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak o opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opage-bre op-agebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak op-agebreak opagebreak opage-bre opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak opagebreak " L = \mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4 $$

  1. Combine the terms into a single fraction. To evaluate the limit, we first combine the expression inside the limit into a single rational function by finding a common denominator, which is (x+1)(x + 1)(x+1). L=lim⁡x→∞(x2+x+1−(ax+b)(x+1)x+1)L = \mathop {\lim }\limits_{x \to \infty } \left( {{{x^2} + x + 1 - (ax + b)(x + 1)} \over {x + 1}} \right)L=x→∞lim​(x+1x2+x+1−(ax+b)(x+1)​) Now, we expand the product in the numerator: (ax+b)(x+1)=ax(x+1)+b(x+1)=ax2+ax+bx+b(ax + b)(x + 1) = ax(x+1) + b(x+1) = ax^2 + ax + bx + b(ax+b)(x+1)=ax(x+1)+b(x+1)=ax2+ax+bx+b Substitute this back into the expression: L=lim⁡x→∞(x2+x+1−(ax2+ax+bx+b)x+1)L = \mathop {\lim }\limits_{x \to \infty } \left( {{{x^2} + x + 1 - (ax^2 + ax + bx + b)} \over {x + 1}} \right)L=x→∞lim​(x+1x2+x+1−(ax2+ax+bx+b)​)

  2. Simplify the numerator. Combine the terms in the numerator by grouping the powers of xxx. L=lim⁡x→∞(x2+x+1−ax2−ax−bx−bx+1)L = \mathop {\lim }\limits_{x \to \infty } \left( {{{x^2} + x + 1 - ax^2 - ax - bx - b} \over {x + 1}} \right)L=x→∞lim​(x+1x2+x+1−ax2−ax−bx−b​) L = \mathop {\lim }\limits_{x \to \infty } \left( {{(1 - a){x^2} + (1 - a - b)x + (1 - b)} \over {x + 1}}} \right)

  3. Analyze the condition for a finite limit. The limit is of a rational function as x→∞x \to \inftyx→∞. For such a limit to exist and be a finite non-zero number, the degree of the numerator must be equal to the degree of the denominator. If the degree of the numerator is greater than the degree of the denominator, the limit will be ±∞\pm\infty±∞. If the degree of the numerator is less than the degree of the denominator, the limit will be 0.

    The degree of the denominator, x+1x+1x+1, is 1. The numerator is (1−a)x2+(1−a−b)x+(1−b)(1 - a)x^2 + (1 - a - b)x + (1 - b)(1−a)x2+(1−a−b)x+(1−b). For the limit to be a finite number (in this case, 4), the degree of the numerator must also be 1. This requires the coefficient of the x2x^2x2 term to be zero. 1−a=01 - a = 01−a=0 a=1a = 1a=1

  4. Substitute the value of 'a' and evaluate the limit. Now that we have a=1a=1a=1, we substitute it back into the limit expression: L = \mathop {\lim }\limits_{x \to \infty } \left( {{(1 - 1){x^2} + (1 - 1 - b)x + (1 - b)} \over {x + 1}}} \right) L=lim⁡x→∞((0)x2+(−b)x+(1−b)x+1)L = \mathop {\lim }\limits_{x \to \infty } \left( {{{(0){x^2} + (-b)x + (1 - b)} \over {x + 1}}} \right)L=x→∞lim​(x+1(0)x2+(−b)x+(1−b)​) L=lim⁡x→∞(−bx+(1−b)x+1)L = \mathop {\lim }\limits_{x \to \infty } \left( {{{ - bx + (1 - b)} \over {x + 1}}} \right)L=x→∞lim​(x+1−bx+(1−b)​) Now, the degree of the numerator is 1 and the degree of the denominator is 1. The limit as x→∞x \to \inftyx→∞ of a rational function with equal degrees in the numerator and denominator is the ratio of their leading coefficients. The leading coefficient of the numerator is −b-b−b. The leading coefficient of the denominator is 111. Therefore, the limit is: L=−b1=−bL = {{-b} \over 1} = -bL=1−b​=−b

  5. Solve for 'b'. We are given that the limit is 4. So, we set our result for the limit equal to 4. L=−b=4L = -b = 4L=−b=4 This gives us: b=−4b = -4b=−4

  6. Conclusion. We have found that a=1a = 1a=1 and b=−4b = -4b=−4.

  7. Match with the options. The calculated values a=1,b=−4a = 1, b = -4a=1,b=−4 correspond to option B.

Verification:

Let's check the result by substituting a=1a = 1a=1 and b=−4b = -4b=−4 into the original expression: lim⁡x→∞(x2+x+1x+1−(1)x−(−4))=lim⁡x→∞(x2+x+1x+1−x+4)\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - (1)x - (-4)} \right) = \mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - x + 4} \right)x→∞lim​(x+1x2+x+1​−(1)x−(−4))=x→∞lim​(x+1x2+x+1​−x+4) = \mathop {\lim }\limits_{x \to \infty } \left( {{{x^2} + x + 1 - x(x+1) + 4(x+1)} \over {x + 1}}} \right) = \mathop {\lim }\limits_{x \to \infty } \left( {{{x^2} + x + 1 - x^2 - x + 4x + 4} \over {x + 1}}} \right) =lim⁡x→∞(4x+5x+1)= \mathop {\lim }\limits_{x \to \infty } \left( {{{4x + 5} \over {x + 1}}} \right)=x→∞lim​(x+14x+5​) Divide the numerator and denominator by xxx: =lim⁡x→∞(4+5/x1+1/x)=4+01+0=4= \mathop {\lim }\limits_{x \to \infty } \left( {{{4 + {5/x}} \over {1 + {1/x}}}} \right) = {{4 + 0} \over {1 + 0}} = 4=x→∞lim​(1+1/x4+5/x​)=1+04+0​=4 The result matches the given value, so our values for aaa and bbb are correct.

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