New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Limits Continuity and Differentiability question

2019 · Shift 2 · Q24

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Limits Continuity and Differentiability
2019 · Shift 2 · Q24

Limits Continuity and Differentiability question

2019 · Shift 2 · Q24

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMultiple correct+4 / −1

Let f : R be a function. We say that f has PROPERTY 1 if

\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}

exists and is finite, and PROPERTY 2 if

\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}

exists and is finite. Then which of the following options is/are correct?

A
f(x) = sin x has PROPERTY 2
B
f(x) = x2/3 has PROPERTY 1
C
f(x) = |x| has PROPERTY 1
D
f(x) = x|x| has PROPERTY 2

More from Limits Continuity and Differentiability

For every twice differentiable function $f : R \to [- 2, 2]$ with $(f (0))^{2} + (f^{'} (0))^{2} = 85$ , which of the following statement(s) is(are) TRUE?2018 · Multiple correct
Let f : R $\to$ R and g : R $\to$ R be two non-constant differentiable functions. If f'(x) = (e(f(x) $-$ g(x))) g'(x) for all x $\in$ R and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?2018 · Multiple correct
The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ....................2018 · Numerical
Let f : (0, $π$ ) $\to$ R be a twice differentiable function such that $t \to x lim \frac{f ( x ) s i n t - f ( t ) s i n x}{t - x} = sin^{2} x$ for all x $\in$ (0, $π$ ). If $f (\frac{π}{6}) = - \frac{π}{12}$ …2018 · Multiple correct
Let $f_{1} : R \to R$ , $f_{2} : (- \frac{π}{2}, \frac{π}{2}) \to R$ , $f_{3} : (- 1, e^{π /2} - 2) \to R$ and $f_{4} : R \to R$ be functions defined by (i) $f_{1} (x) = sin (1 - e^{- x^{2}})$ …2018 · MCQ
Let f : R $\to$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?2017 · Multiple correct
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $f (x) = x cos (π (x + [x]))$ is discontinuous?2017 · Multiple correct
Let f : R $\to$ R be a differentiable function such that f(0) = 0, $f (\frac{π}{2}) = 3$ and f'(0) = 1. If $g (x) = x \int π /2 [f^{'} (t) cosec t - cot t cosec t f (t)] d t$ for $x \in (0, \frac{π}{2}]$ …2017 · Numerical