Limits, Continuity and Differentiability PYQs - Last 10 Years
WB JEE / Mathematics / Calculus / 57 recent questions
MathematicsCalculus2017-2026
Practice 57 WB JEE Mathematics questions from Limits, Continuity and Differentiability. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.
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Last 10 Years Limits, Continuity and Differentiability Questions
Showing 50 of 57 filtered questions.
1Limits Continuity And Differentiability
If $a=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x=n \pi)$ and $b=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x \neq n \pi)$, then numerical value of the area of the triangle whose vertices are (a, b), (-2, 1) and (2, ...
MCQ+1 / -0.252026
2Limits Continuity And Differentiability
For a real number $y$, consider $(y)$ denotes the greatest integer less than or equal to $y$. If $f(x)=\frac{\tan (\pi[x-\pi])}{1+[x]^2}$, then
MCQ+1 / -0.252026
3Limits Continuity And Differentiability
If $f(x)$ is differentiable for all $x \in \mathbb{R}$ and satisfies the relation
$x=\mathop {\lim }\limits_{n \to \infty }\frac{\left[1^2(f(x))^x\right]+\left[2^2(f(x))^x\right]+\ldots+\left[n^2(f(x))^x\right]}{n^3}$ where [.] denotes the ...
$x=\mathop {\lim }\limits_{n \to \infty }\frac{\left[1^2(f(x))^x\right]+\left[2^2(f(x))^x\right]+\ldots+\left[n^2(f(x))^x\right]}{n^3}$ where [.] denotes the ...
MCQ+2 / -02026
4Limits Continuity And Differentiability
Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\mathop {\lim }\limits_{x \to a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m(\neq 0)$, then the value of $\lambda$ ix
MCQ+1 / -0.252026
5Limits Continuity And Differentiability
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
MCQ+2 / -0.52025
6Limits Continuity And Differentiability
Let $f:[0,1] \rightarrow \mathbb{R}$ and $g:[0,1] \rightarrow \mathbb{R}$ be defined as follows :
$\left.\begin{array}{rl}f(x) & =1 \text { if } x \text { is rational } \\ & =0 \text { if } x \text { is irrational }\end{array}\right]$ and
$...
$\left.\begin{array}{rl}f(x) & =1 \text { if } x \text { is rational } \\ & =0 \text { if } x \text { is irrational }\end{array}\right]$ and
$...
MCQM+2 / -02025
7Limits Continuity And Differentiability
A function $f$ is defined by $f(x)=2+(x-1)^{2 / 3}$ on $[0,2]$. Which of the following statements is incorrect?
MCQ+1 / -0.252025
8Limits Continuity And Differentiability
The set of points of discontinuity of the function $f(x)=x-[x], x \in \mathbb{R}$ is
MCQ+1 / -0.252025
9Limits Continuity And Differentiability
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
If the function ' $f$ ' is differentiable at $x=0$, then $f$ is
MCQ+1 / -0.252025
10Limits Continuity And Differentiability
$\lim\limits_{x \rightarrow 0} \frac{\tan \left(\left[-\pi^2\right] x^2\right)-x^2 \tan \left(\left[-\pi^2\right]\right)}{\sin ^2 x}$ equals
MCQ+1 / -0.252025
11Limits Continuity And Differentiability
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 $[0,5]$
MCQ+1 / -0.252025
12Limits Continuity And Differentiability
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
MCQ+2 / -0.52025
13Limits Continuity And Differentiability
Let $f(x)=|1-2 x|$, then
MCQ+1 / -0.252025
14Limits Continuity And Differentiability
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 :
MCQ+1 / -0.252025
15Limits Continuity And Differentiability
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
MCQ+1 / -0.252024
16Limits Continuity And Differentiability
$$
\text { Let } f(x)=\left|\begin{array}{ccc}
\cos x & x & 1 \\
2 \sin x & x^3 & 2 x \\
\tan x & x & 1
\end{array}\right| \text {, then } \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}=
$$
MCQ+1 / -0.252024
17Limits Continuity And Differentiability
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...
MCQ+2 / -0.52023
18Limits Continuity And Differentiability
Let \(f(x) = [{x^2}]\sin \pi x,x > 0\). Then
MCQ+1 / -0.252023
19Limits Continuity And Differentiability
Let \(f(x) = \left\{ {\matrix{
{x + 1,} & { - 1 \le x \le 0} \cr
{ - x,} & {0 < x \le 1} \cr
} } \right.\)
MCQ+1 / -0.252023
20Limits Continuity And Differentiability
f(x) is a differentiable function and given \(f'(2) = 6\) and \(f'(1) = 4\), then \(L = \mathop {\lim }\limits_{h \to 0} {{f(2 + 2h + {h^2}) - f(2)} \over {f(1 + h - {h^2}) - f(1)}}\)
MCQ+1 / -0.252023
21Limits Continuity And Differentiability
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
MCQ+1 / -0.252023
22Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to \infty } \left\{ {x - \root n \of {(x - {a_1})(x - {a_2})\,...\,(x - {a_n})} } \right\}\) where \({a_1},{a_2},\,...,\,{a_n}\) are positive rational numbers. The limit
MCQ+1 / -0.252023
23Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + 1} \over {x + 1}} - ax - b} \right),(a,b \in R)\) = 0. Then
MCQ+2 / -0.52022
24Limits Continuity And Differentiability
Let f : [a, b] \(\to\) R be continuous in [a, b], differentiable in (a, b) and f(a) = 0 = f(b). Then
MCQ+1 / -0.252022
25Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x}\ln \sqrt {{{1 + x} \over {1 - x}}} } \right)\) is
MCQ+1 / -0.252022
26Limits Continuity And Differentiability
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
MCQ+1 / -0.252022
27Limits Continuity And Differentiability
The values of a, b, c for which the function $$f(x) = \left\{ \matrix{
{{\sin (a + 1)x + \sin x} \over x},x < 0 \hfill \cr
c,x = 0 \hfill \cr
{{{{(x + b{x^2})}^{{1 \over 2}}} - {x^{{1 \over 2}}}} \over {b{x^{{1 \over 2}}}}},x > 0 \h...
{{\sin (a + 1)x + \sin x} \over x},x < 0 \hfill \cr
c,x = 0 \hfill \cr
{{{{(x + b{x^2})}^{{1 \over 2}}} - {x^{{1 \over 2}}}} \over {b{x^{{1 \over 2}}}}},x > 0 \h...
MCQ+1 / -0.252022
28Limits Continuity And Differentiability
$$\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt n } \over {\sqrt {{n^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 4)}^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 8)}^3}} }} + .... + {{\sqrt n } \over {\sqrt {{{[n + 4(n - 1)]}^3}} }}} \...
MCQM+2 / -02021
29Limits Continuity And Differentiability
The \(\mathop {\lim }\limits_{x \to \infty } {\left( {{{3x - 1} \over {3x + 1}}} \right)^{4x}}\) equals
MCQ+2 / -0.52021
30Limits Continuity And Differentiability
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,
MCQ+1 / -0.252021
31Limits Continuity And Differentiability
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
MCQ+1 / -0.252021
32Limits Continuity And Differentiability
If \(I = \mathop {\lim }\limits_{x \to 0} sin\left( {{{{e^x} - x - 1 - {{{x^2}} \over 2}} \over {{x^2}}}} \right)\), then limit
MCQ+1 / -0.252021
33Limits Continuity And Differentiability
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
MCQ+1 / -0.252020
34Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)\)
MCQ+2 / -0.52020
35Limits Continuity And Differentiability
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
MCQ+2 / -0.52020
36Limits Continuity And Differentiability
Let \(f(x) = {1 \over 3}x\sin x - (1 - \cos \,x)\). The smallest positive integer k such that \(\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^k}}} \ne 0\) is
MCQM+2 / -02020
37Limits Continuity And Differentiability
Let \(\phi (x) = f(x) + f(1 - x)\) and \(f(x) < 0\) in [0, 1], then
MCQ+1 / -0.252020
38Limits Continuity And Differentiability
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
MCQ+1 / -0.252020
39Limits Continuity And Differentiability
Consider the function \(f(x) = {{{x^3}} \over 4} - \sin \pi x + 3\)
MCQM+2 / -02019
40Limits Continuity And Differentiability
Let \(f:[1,3] \to R\) be a continuous function that is differentiable in (1, 3) an f'(x) = | f(x) |2 + 4 for all x\(\in\) (1, 3). Then,
MCQM+1 / -0.252019
41Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0\)
MCQ+1 / -0.252019
42Limits Continuity And Differentiability
The value of \(\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]\) is
MCQ+2 / -0.52019
43Limits Continuity And Differentiability
The limit of the interior angle of a regular polygon of n sides as n \(\to\) \(\infty\) is
MCQ+1 / -0.252019
44Limits Continuity And Differentiability
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 mov...
MCQ+2 / -0.52019
45Limits Continuity And Differentiability
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
MCQ+2 / -0.52019
46Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to {0^ + }} {({e^x} + x)^{1/x}}\)
MCQ+1 / -0.252019
47Limits Continuity And Differentiability
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,
MCQ+2 / -0.52018
48Limits Continuity And Differentiability
Let f : R \(\to\) R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then
MCQ+1 / -0.252018
49Limits Continuity And Differentiability
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
MCQ+1 / -0.252018
50Limits Continuity And Differentiability
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
MCQ+1 / -0.252018