Limits, Continuity and Differentiability · Mathematics · WB JEE

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MCQ (Single Correct Answer)

1

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{{a^{\cot x}} - {a^{\cos x}}} \over {\cot x - \cos x}},a > 0$$

WB JEE 2008
2

Rolle's theorem is not applicable to the function $$f(x) = |x|$$ for $$ - 2 \le x \le 2$$ because

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3

The value of the limit $$\mathop {\lim }\limits_{x \to 2} {{{e^{3x - 6}} - 1} \over {\sin (2 - x)}}$$ is

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4

The $$\mathop {\lim }\limits_{x \to 2} {5 \over {\sqrt 2 - \sqrt x }}$$ is

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5

A function f(x) is defined as follows for real x

$$f(x) = \left\{ {\matrix{ {1 - {x^2}} & , & {for\,x < 1} \cr 0 & , & {for\,x = 1} \cr {1 + {x^2}} & , & {for\,x > 1} \cr } } \right.$$

Then

WB JEE 2008
6

Let $$f(x) = {{\sqrt {x + 3} } \over {x + 1}}$$, then the value of $$\mathop {\lim }\limits_{x \to - 3 - 0} f(x)$$ is

WB JEE 2009
7

$$f(x) = x + |x|$$ is continuous for

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8

The value of $$\mathop {\lim }\limits_{x \to 1} {{\sin ({e^{x - 1}} - 1)} \over {\log x}}$$ is

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9

The value of $$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x + \cos x - 1} \over {{x^2}}}$$ is

WB JEE 2010
10

The value of $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 5{x^2}} \over {1 + 3{x^2}}}} \right)^{{1 \over {{x^2}}}}}$$ is

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11

If $$f(5) = 7$$ and $$f'(5) = 7$$, then $$\mathop {\lim }\limits_{x \to 5} {{x\,f(5) - 5f(x)} \over {x - 5}}$$ is given by

WB JEE 2010
12

If $$y = (1 + x)(1 + {x^2})(1 + {x^4})\,.....\,(1 + {x^{2n}})$$, then the value of $${\left( {{{dy} \over {dx}}} \right)_{x = 0}}$$ is

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13

The value of f(0) so that the function $$f(x) = {{1 - \cos (1 - \cos x)} \over {{x^4}}}$$ is continuous everywhere is

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14
$$\mathop {\lim }\limits_{x \to 0} {{\sin |x|} \over x}$$ is equal to
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15

In which of the following functions, Rolle's theorem is applicable?

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16

The value of $$\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + ..... + {x^n} - n} \over {x - 1}}$$ is

WB JEE 2011
17

$$\mathop {\lim }\limits_{x \to 0} {{\sin (\pi {{\sin }^2}x)} \over {{x^2}}} = $$

WB JEE 2011
18

If the function $$f(x) = \left\{ {\matrix{ {{{{x^2} - (A + 2)x + A} \over {x - 2}},} & {for\,x \ne 2} \cr {2,} & {for\,x = 2} \cr } } \right.$$ is continuous at x = 2, then

WB JEE 2011
19

$$f(x) = \left\{ {\matrix{ {[x] + [ - x],} & {when\,x \ne 2} \cr {\lambda ,} & {when\,x = 0} \cr } } \right.$$

If f(x) is continuous at x = 2, the value of $$\lambda$$ will be

WB JEE 2011
20

For the function $$f(x) = {e^{\cos x}}$$, Rolle's theorem is

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21

$$f(x) = \left\{ {\matrix{ {0,} & {x = 0} \cr {x - 3,} & {x > 0} \cr } } \right.$$

The function f(x) is

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22

The function f(x) = ax + b is strictly increasing for all real x if

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23
If $f(x)=\left\{\begin{array}{ll}x^2+3 x+a, & x \leq 1 \\ b x+2, & x>1\end{array}, x \in \mathbb{R}\right.$, is everywhere differentiable, then :
WB JEE 2025
24

Let $f(x)=|1-2 x|$, then

WB JEE 2025
25

A function $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfies $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3}$ for all $x, y \in \mathbb{R}$. If the function ' $f$ ' is differentiable at $x=0$, then $f$ is

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26

The set of points of discontinuity of the function $f(x)=x-[x], x \in \mathbb{R}$ is

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27

A function $f$ is defined by $f(x)=2+(x-1)^{2 / 3}$ on $[0,2]$. Which of the following statements is incorrect?

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28

Let $f(x)$ be continuous on $[0,5]$ and differentiable in $(0,5)$. If $f(0)=0$ and $\left|f^{\prime}(x)\right| \leq \frac{1}{5}$ for all $x$ in $(0,5)$, then $\forall x$ in $[0,5]$

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29

$\lim\limits_{x \rightarrow 0} \frac{\tan \left(\left[-\pi^2\right] x^2\right)-x^2 \tan \left(\left[-\pi^2\right]\right)}{\sin ^2 x}$ equals

WB JEE 2025
30

Let $f(x)=|x-\alpha|+|x-\beta|$, where $\alpha, \beta$ are the roots of the equation $x^2-3 x+2=0$. Then the number of points in $[\alpha,\beta]$ at which $f$ is not differentiable is

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31

Let $a_n$ denote the term independent of $x$ in the expansion of $\left[x+\frac{\sin (1 / n)}{x^2}\right]^{3 n}$, then $\lim \limits_{n \rightarrow \infty} \frac{\left(a_n\right) n!}{{ }^{3 n} P_n}$ equals

WB JEE 2025
32

$$ \text { Let } f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x^3 & 2 x \\ \tan x & x & 1 \end{array}\right| \text {, then } \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}= $$

WB JEE 2024
33

If $$\alpha, \beta$$ are the roots of the equation $$a x^2+b x+c=0$$ then $$\lim _\limits{x \rightarrow \beta} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\beta)^2}$$ is

WB JEE 2024
34

$$\mathop {\lim }\limits_{x \to \infty } \left\{ {x - \root n \of {(x - {a_1})(x - {a_2})\,...\,(x - {a_n})} } \right\}$$ where $${a_1},{a_2},\,...,\,{a_n}$$ are positive rational numbers. The limit

WB JEE 2023
35

Let $$f:[1,3] \to R$$ be continuous and be derivable in (1, 3) and $$f'(x) = {[f(x)]^2} + 4\forall x \in (1,3)$$. Then

WB JEE 2023
36

f(x) is a differentiable function and given $$f'(2) = 6$$ and $$f'(1) = 4$$, then $$L = \mathop {\lim }\limits_{h \to 0} {{f(2 + 2h + {h^2}) - f(2)} \over {f(1 + h - {h^2}) - f(1)}}$$

WB JEE 2023
37

Let $$f(x) = \left\{ {\matrix{ {x + 1,} & { - 1 \le x \le 0} \cr { - x,} & {0 < x \le 1} \cr } } \right.$$

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38

Let $$f(x) = [{x^2}]\sin \pi x,x > 0$$. Then

WB JEE 2023
39

The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {\left( {{1 \over {2\,.\,3}} + {1 \over {{2^2}\,.\,3}}} \right) + \left( {{1 \over {{2^2}\,.\,{3^2}}} + {1 \over {{2^3}\,.\,{3^2}}}} \right)\, + \,...\, + \,\left( {{2 \over {{2^n}\,.\,{3^n}}} + {1 \over {{2^{n + 1}}\,.\,3n}}} \right)} \right]$$ is

WB JEE 2023
40

The values of a, b, c for which the function $$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin x} \over x},x < 0 \hfill \cr c,x = 0 \hfill \cr {{{{(x + b{x^2})}^{{1 \over 2}}} - {x^{{1 \over 2}}}} \over {b{x^{{1 \over 2}}}}},x > 0 \hfill \cr} \right.$$ is continuous at x = 0, are

WB JEE 2022
41

Let $$f(x) = {a_0} + {a_1}|x| + {a_2}|x{|^2} + {a_3}|x{|^3}$$, where $${a_0},{a_1},{a_2},{a_3}$$ are real constants. Then f(x) is differentiable at x = 0

WB JEE 2022
42

$$\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x}\ln \sqrt {{{1 + x} \over {1 - x}}} } \right)$$ is

WB JEE 2022
43

Let f : [a, b] $$\to$$ R be continuous in [a, b], differentiable in (a, b) and f(a) = 0 = f(b). Then

WB JEE 2022
44

$$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + 1} \over {x + 1}} - ax - b} \right),(a,b \in R)$$ = 0. Then

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45
If $$I = \mathop {\lim }\limits_{x \to 0} sin\left( {{{{e^x} - x - 1 - {{{x^2}} \over 2}} \over {{x^2}}}} \right)$$, then limit
WB JEE 2021
46
Let $${S_n} = {\cot ^{ - 1}}2 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18 + {\cot ^{ - 1}}32 + ....$$ to nth term. Then $$\mathop {\lim }\limits_{n \to \infty } {S_n}$$ is
WB JEE 2021
47
Let f : D $$\to$$ R where D = [$$-$$0, 1] $$\cup$$ [2, 4] be defined by

$$f(x) = \left\{ {\matrix{ {x,} & {if} & {x \in [0,1]} \cr {4 - x,} & {if} & {x \in [2,4]} \cr } } \right.$$ Then,
WB JEE 2021
48
The $$\mathop {\lim }\limits_{x \to \infty } {\left( {{{3x - 1} \over {3x + 1}}} \right)^{4x}}$$ equals
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49
Let $$\phi (x) = f(x) + f(1 - x)$$ and $$f(x) < 0$$ in [0, 1], then
WB JEE 2020
50
If $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + cx} \over {1 - cx}}} \right)^{{1 \over x}}} = 4$$, then $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 2cx} \over {1 - 2cx}}} \right)^{{1 \over x}}}$$ is
WB JEE 2020
51
Let f : R $$ \to $$ R be twice continuously differentiable (or f" exists and is continuous) such that f(0) = f(1) = f'(0) = 0. Then
WB JEE 2020
52
Let $$0 < \alpha < \beta < 1$$. Then, $$\mathop {\lim }\limits_{n \to \infty } \int\limits_{1/(k + \beta )}^{1/(k + \alpha )} {{{dx} \over {1 + x}}} $$ is
WB JEE 2020
53
$$\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)$$
WB JEE 2020
54
$$\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0$$
WB JEE 2019
55
The limit of the interior angle of a regular polygon of n sides as n $$ \to $$ $$\infty $$ is
WB JEE 2019
56
$$\mathop {\lim }\limits_{x \to {0^ + }} {({e^x} + x)^{1/x}}$$
WB JEE 2019
57
Let $$a = \min \{ {x^2} + 2x + 3:x \in R\} $$ and $$b = \mathop {\lim }\limits_{\theta \to 0} {{1 - \cos \theta } \over {{\theta ^2}}}$$. Then $$\sum\limits_{r = 0}^n {{a^r}{b^{n - r}}} $$ is
WB JEE 2019
58
A particle starts at the origin and moves 1 unit horizontally to the right and reaches P1, then it moves $${1 \over 2}$$ unit vertically up and reaches P2, then it moves $${1 \over 4}$$ unit horizontally to right and reaches P3, then it moves $${1 \over 8}$$ unit vertically down and reaches P4, then it moves $${1 \over 16}$$ unit horizontally to right and reaches P5 and so on. Let Pn = (xn, yn) and $$\mathop {\lim }\limits_{n \to \infty } {x_n} = \alpha $$ and $$\mathop {\lim }\limits_{n \to \infty } {y_n} = \beta $$. Then, ($$\alpha$$, $$\beta$$) is
WB JEE 2019
59
The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]$$ is
WB JEE 2019
60
Let f : [a, b] $$ \to $$ R be differentiable on [a, b] and k $$ \in $$ R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then
WB JEE 2018
61
Let $$f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7$$.

Then $$\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}$$
WB JEE 2018
62
Let f : [a, b] $$ \to $$ R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then
WB JEE 2018
63
Let f : R $$ \to $$ R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then
WB JEE 2018
64
Let $$f(x) = \left\{ {\matrix{ { - 2\sin x,} & {if\,x \le - {\pi \over 2}} \cr {A\sin x + B,} & {if\, - {\pi \over 2} < x < {\pi \over 2}} \cr {\cos x} & {if\,x \ge {\pi \over 2}} \cr } } \right.$$. Then,
WB JEE 2018
65
Consider the non-constant differentiable function f one one variable which obeys the relation $${{f(x)} \over {f(y)}} = f(x - y)$$. If f' (0) = p and f' (5) = q, then f' ($$-$$5) is
WB JEE 2017
66
If f'' (0) = k, k $$ \ne $$ 0, then the value of

$$\mathop {\lim }\limits_{x \to 0} {{2f(x) - 3f(2x) + f(4x)} \over {{x^2}}}$$ is
WB JEE 2017
67
Let $$f(x) = \left\{ {\matrix{ {{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr {0,} & {if\,x = 0} \cr } } \right.$$, $$(p,q \in R)$$. Then, Lagrange's mean value theorem is applicable to f(x) in closed interval [0, x]
WB JEE 2017
68
$$\mathop {\lim }\limits_{x \to 0} {(\sin x)^{2\tan x}}$$ is equal to
WB JEE 2017
69
Let for all x > 0, $$f(x) = \mathop {\lim }\limits_{n \to \infty } n({x^{1/n}} - 1)$$, then
WB JEE 2017
70
If $$y = (1 + x)(1 + {x^2})(1 + {x^4})...(1 + {x^{2n}})$$, then the value of $$\left( {{{dy} \over {dx}}} \right)$$ at x = 0 is
WB JEE 2016
71
$$\mathop {\lim }\limits_{x \to 1} {\left( {{{1 + x} \over {2 + x}}} \right)^{{{(1 - \sqrt x )} \over {(1 - x)}}}}$$ is equal to
WB JEE 2016
72
The value of

$$\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt {n + 1} + \sqrt {n + 2} + ... + \sqrt {2n - 1} } \over {{n^{3/2}}}}} \right\}$$ is
WB JEE 2016

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