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Limits Continuity and Differentiability question

2021 · 27 Aug · Shift 2 · Q35
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  5. /2021 · 27 Aug · Shift 2 · Q35

Limits Continuity and Differentiability question

2021 · 27 Aug · Shift 2 · Q35

JEE MainMathematicsLimits Continuity and DifferentiabilityMCQ+4 / −1
If lim⁡x→∞(x2−x+1−ax)=b\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = bx→∞lim​(x2−x+1​−ax)=b, then the ordered pair (a, b) is :
  1. A
    (1,12)\left( {1,{1 \over 2}} \right)(1,21​)
  2. B
    (1,−12)\left( {1, - {1 \over 2}} \right)(1,−21​)
  3. C
    (−1,12)\left( { - 1,{1 \over 2}} \right)(−1,21​)
  4. D
    (−1,−12)\left( { - 1, - {1 \over 2}} \right)(−1,−21​)
View written solutionFree

Correct answer: B

  1. We need to evaluate
lim⁡x→∞(x2−x+1−ax)=b.\lim_{x\to\infty}\left(\sqrt{x^2-x+1}-ax\right)=b.x→∞lim​(x2−x+1​−ax)=b.

We must choose aaa so that the limit is finite, and then compute bbb.

  1. For large xxx,
x2−x+1=x1−1x+1x2.\sqrt{x^2-x+1}=x\sqrt{1-\frac1x+\frac1{x^2}}.x2−x+1​=x1−x1​+x21​​.

So the expression becomes

x1−1x+1x2−ax.x\sqrt{1-\frac1x+\frac1{x^2}}-ax.x1−x1​+x21​​−ax.
  1. As x→∞x\to\inftyx→∞,
1−1x+1x2→1.\sqrt{1-\frac1x+\frac1{x^2}}\to 1.1−x1​+x21​​→1.

Hence the dominant term is approximately x(1−a)x(1-a)x(1−a).

For the limit to be finite, we must have

1−a=0  ⟹  a=1.1-a=0 \implies a=1.1−a=0⟹a=1.
  1. Now substitute a=1a=1a=1:
b=lim⁡x→∞(x2−x+1−x).b=\lim_{x\to\infty}\left(\sqrt{x^2-x+1}-x\right).b=x→∞lim​(x2−x+1​−x).

Rationalize:

x2−x+1−x=(x2−x+1)−x2x2−x+1+x=−x+1x2−x+1+x.\sqrt{x^2-x+1}-x =\frac{(x^2-x+1)-x^2}{\sqrt{x^2-x+1}+x} =\frac{-x+1}{\sqrt{x^2-x+1}+x}.x2−x+1​−x=x2−x+1​+x(x2−x+1)−x2​=x2−x+1​+x−x+1​.
  1. Divide numerator and denominator by xxx:
−x+1x2−x+1+x=−1+1x1−1x+1x2+1.\frac{-x+1}{\sqrt{x^2-x+1}+x} =\frac{-1+\frac1x}{\sqrt{1-\frac1x+\frac1{x^2}}+1}.x2−x+1​+x−x+1​=1−x1​+x21​​+1−1+x1​​.

Now let x→∞x\to\inftyx→∞:

b=−11+1=−12.b=\frac{-1}{1+1}=-\frac12.b=1+1−1​=−21​.
  1. Therefore,
(a,b)=(1,−12).(a,b)=\left(1,-\frac12\right).(a,b)=(1,−21​).

This matches option B.

  1. Comparison with stored correct answer: Stored correct answer = B. Our derived answer = B. So they agree.
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