Limits, Continuity and Differentiability PYQs - Last 10 Years
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MathematicsCalculus2017-2026
Practice 241 JEE Main 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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128 in last 5 years241 in last 10 years
Last 10 Years Limits, Continuity and Differentiability Questions
Showing 50 of 241 filtered questions.
1Limits Continuity And Differentiability
The number of points, at which the function f(x) = | 2x + 1 | \(-\) 3| x + 2 | + | x2 + x \(-\) 2 |, x\(\in\)R is not differentiable, is __________.
INTEGER+4 / -12021
2Limits Continuity And Differentiability
A function f is defined on [\(-\)3, 3] as\(f(x) = \left\{ {\matrix{
{\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le 2} \cr
{[|x|],} & {2 < |x| \le 3} \cr
} } \right.\) where [x] denotes the greatest integer \(\le\) x. The number...
INTEGER+4 / -12021
3Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}\) exists and is equal to b, then the value of a \(-\) 2b is __________.
INTEGER+4 / -12021
4Limits Continuity And Differentiability
If f : R \(\to\) R is a function defined by f(x)= [x - 1] \(\cos \left( {{{2x - 1} \over 2}} \right)\pi\), where [.] denotes the greatest
integer function, then f is :
integer function, then f is :
MCQ+4 / -12021
5Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + {r^2}}}} \right)} } \right\}\) is equal to ______.
INTEGER+4 / -12021
6Limits Continuity And Differentiability
Let f : R \(\to\) R be a function defined as \(f(x) = \left\{ {\matrix{
{3\left( {1 - {{|x|} \over 2}} \right)} & {if} & {|x|\, \le 2} \cr
0 & {if} & {|x|\, > 2} \cr
} } \right.\)Let g : R \(\to\) R be given by $$g(x) = f(x + 2)...
INTEGER+4 / -12021
7Limits Continuity And Differentiability
Let f : R \(\to\) R be defined as $$f(x) = \left\{ {\matrix{
{{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr
} } ...
{{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr
} } ...
MCQ+4 / -12021
8Limits Continuity And Differentiability
If the value of \(\mathop {\lim }\limits_{x \to 0} {(2 - \cos x\sqrt {\cos 2x} )^{\left( {{{x + 2} \over {{x^2}}}} \right)}}\) is equal to ea, then a is equal to __________.
INTEGER+4 / -12021
9Limits Continuity And Differentiability
Let a function f : R \(\to\) R be defined as \(f(x) = \left\{ {\matrix{
{\sin x - {e^x}} & {if} & {x \le 0} \cr
{a + [ - x]} & {if} & {0 < x < 1} \cr
{2x - b} & {if} & {x \ge 1} \cr
} } \right.\) where [ x ] is the greatest...
MCQ+4 / -12021
10Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 0} {{\alpha x{e^x} - \beta {{\log }_e}(1 + x) + \gamma {x^2}{e^{ - x}}} \over {x{{\sin }^2}x}} = 10,\alpha ,\beta ,\gamma \in R\), then the value of \(\alpha\) + \(\beta\) + \(\gamma\) is _____________.
INTEGER+4 / -12021
11Limits Continuity And Differentiability
Let a function g : [ 0, 4 ] \(\to\) R be defined as \(g(x) = \left\{ {\matrix{
{\mathop {\max }\limits_{0 \le t \le x} \{ {t^3} - 6{t^2} + 9t - 3),} & {0 \le x \le 3} \cr
{4 - x,} & {3 < x \le 4} \cr
} } \right.\), then the numb...
INTEGER+4 / -12021
12Limits Continuity And Differentiability
If \(f:R \to R\) is given by \(f(x) = x + 1\), then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \...
MCQ+4 / -12021
13Limits Continuity And Differentiability
Let [t] denote the greatest integer \(\le\) t. The number of points where the function \(f(x) = [x]\left| {{x^2} - 1} \right| + \sin \left( {{\pi \over {[x] + 3}}} \right) - [x + 1],x \in ( - 2,2)\) is not continuous is _____________.
INTEGER+4 / -12021
14Limits Continuity And Differentiability
Let \(f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3\), x \(\in\) R. Then the natural number n for which \(\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44\) is __________.
INTEGER+4 / -12021
15Limits Continuity And Differentiability
If \(f(x) = \left\{ {\matrix{
{{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr
{a{x^2} + b} & {;\,|x|\, < 1} \cr
} } \right.\) is differentiable at every point of the domain, then the values of a and b are respectively :
MCQ+4 / -12021
16Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}}\) is equal to L, then the value of (6L + 1) is
MCQ+4 / -12021
17Limits Continuity And Differentiability
Let f : R \(\to\) R be a function defined as$$f(x) = \left\{ \matrix{
{{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr
b,\,if\,x\, = 0 \hfill \cr
{{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill ...
{{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr
b,\,if\,x\, = 0 \hfill \cr
{{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill ...
MCQ+4 / -12021
18Limits Continuity And Differentiability
Let f : R \(\to\) R satisfy the equation f(x + y) = f(x) . f(y) for all x, y \(\in\)R and f(x) \(\ne\) 0 for any x\(\in\)R. If the function f is differentiable at x = 0 and f'(0) = 3, then $$\mathop {\lim }\limits_{h \to 0} {1 \over h}(f(...
INTEGER+4 / -12021
19Limits Continuity And Differentiability
The value of \(\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}\), where [ x ] denotes the greatest integer \(\le\) x is :
MCQ+4 / -12021
20Limits Continuity And Differentiability
If the function \(f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}\) is continuous at each point in its domain and \(f(0) = {1 \over k}\), then k is ____________.
INTEGER+4 / -12021
21Limits Continuity And Differentiability
The value of \(\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}\), where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :
MCQ+4 / -12021
22Limits Continuity And Differentiability
The value of the limit \(\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}\) is equal to :
MCQ+4 / -12021
23Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 0} {{a{e^x} - b\cos x + c{e^{ - x}}} \over {x\sin x}} = 2\), then a + b + c is equal to ____________.
INTEGER+4 / -12021
24Limits Continuity And Differentiability
Let the functions f : R \(\to\) R and g : R \(\to\) R be defined as :\(f(x) = \left\{ {\matrix{
{x + 2,} & {x < 0} \cr
{{x^2},} & {x \ge 0} \cr
} } \right.\) and $$g(x) = \left\{ {\matrix{
{{x^3},} & {x < 1} \cr
{3x ...
{{x^3},} & {x < 1} \cr
{3x ...
MCQ+4 / -12021
25Limits Continuity And Differentiability
Let \({S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)}\). Then \(\mathop {\lim }\limits_{k \to \infty } {S_k}\) is equal to :
MCQ+4 / -12021
26Limits Continuity And Differentiability
Let f : R \(\to\) R and g : R \(\to\) R be defined as \(f(x) = \left\{ {\matrix{
{x + a,} & {x < 0} \cr
{|x - 1|,} & {x \ge 0} \cr
} } \right.\) and $$g(x) = \left\{ {\matrix{
{x + 1,} & {x < 0} \cr
{{{(x - 1)}^2} + ...
{x + 1,} & {x < 0} \cr
{{{(x - 1)}^2} + ...
INTEGER+4 / -12021
27Limits Continuity And Differentiability
Let \(\alpha\) \(\in\) R be such that the function $$f(x) = \left\{ {\matrix{
{{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr
} } \...
{{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr
} } \...
MCQ+4 / -12021
28Limits Continuity And Differentiability
Let f : S \(\to\) S where S = (0, \(\infty\)) be a twice differentiable function such that f(x + 1) = xf(x). If g : S \(\to\) R be defined as g(x) = loge f(x), then the value of |g''(5) \(-\) g''(1)| is equal to :
MCQ+4 / -12021
29Limits Continuity And Differentiability
If $$f(x) = \left\{ {\matrix{
{{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr
{b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr
{{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \ov...
{{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr
{b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr
{{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \ov...
MCQ+4 / -12020
30Limits Continuity And Differentiability
Let ƒ be any function continuous on [a, b] and
twice differentiable on (a, b). If for all x \(\in\) (a, b),
ƒ'(x) > 0 and ƒ''(x) < 0, then for any c \(\in\) (a, b),
\({{f(c) - f(a)} \over {f(b) - f(c)}}\) is greater than :
twice differentiable on (a, b). If for all x \(\in\) (a, b),
ƒ'(x) > 0 and ƒ''(x) < 0, then for any c \(\in\) (a, b),
\({{f(c) - f(a)} \over {f(b) - f(c)}}\) is greater than :
MCQ+4 / -12020
31Limits Continuity And Differentiability
Let [t] denote the greatest integer \(\le\) t
and \(\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A\). Then the function,
f(x) = [x2]sin(\(\pi\)x) is discontinuous, when x is
equal to :
and \(\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A\). Then the function,
f(x) = [x2]sin(\(\pi\)x) is discontinuous, when x is
equal to :
MCQ+4 / -12020
32Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}\) is equal to
MCQ+4 / -12020
33Limits Continuity And Differentiability
Let S be the set of all functions ƒ : [0,1] \(\to\) R,
which are continuous on [0,1] and differentiable
on (0,1). Then for every ƒ in S, there exists a
c \(\in\) (0,1), depending on ƒ, such that
which are continuous on [0,1] and differentiable
on (0,1). Then for every ƒ in S, there exists a
c \(\in\) (0,1), depending on ƒ, such that
MCQ+4 / -12020
34Limits Continuity And Differentiability
Let S be the set of points where the function, ƒ(x) = |2-|x-3||, x \(\in\) R is not differentiable. Then \(\sum\limits_{x \in S} {f(f(x))}\) is equal to_____.
INTEGER+4 / -02020
35Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 2} {{{3^x} + {3^{3 - x}} - 12} \over {{3^{ - x/2}} - {3^{1 - x}}}}\) is equal to_______.
INTEGER+4 / -02020
36Limits Continuity And Differentiability
If the function ƒ defined on \(\left( { - {1 \over 3},{1 \over 3}} \right)\) by
f(x) = $$\left\{ {\matrix{
{{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr
{k,} & {when\,x = 0} \cr
} } \r...
f(x) = $$\left\{ {\matrix{
{{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr
{k,} & {when\,x = 0} \cr
} } \r...
INTEGER+4 / -02020
37Limits Continuity And Differentiability
Let f : R \(\to\) R be defined as
$$f\left( x \right) = \left\{ {\matrix{
{{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \cr
{0,} & {x = 0} \cr
{{x^5}\cos \left( {{1 \over x}} \right) + \lambda {x^2},} & {x > 0}...
$$f\left( x \right) = \left\{ {\matrix{
{{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \cr
{0,} & {x = 0} \cr
{{x^5}\cos \left( {{1 \over x}} \right) + \lambda {x^2},} & {x > 0}...
INTEGER+4 / -02020
38Limits Continuity And Differentiability
For all twice differentiable functions f : R \(\to\) R,
with f(0) = f(1) = f'(0) = 0
with f(0) = f(1) = f'(0) = 0
MCQ+4 / -12020
39Limits Continuity And Differentiability
Let f : R \(\to\) R be a function defined by f(x) = max {x, x2}. Let S denote the set of all points in R, where f is not differentiable.
Then :
Then :
MCQ+4 / -12020
40Limits Continuity And Differentiability
If the function \(f\left( x \right) = \left\{ {\matrix{
{{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr
{{k_2}\cos x,} & {x > \pi } \cr
} } \right.\) is twice differentiable, then the ordered pair (k1, k2) is equ...
MCQ+4 / -12020
41Limits Continuity And Differentiability
If \(\alpha\) is positive root of the equation, p(x) = x2 - x - 2 = 0, then
\(\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}\) is equal to :
\(\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}\) is equal to :
MCQ+4 / -12020
42Limits Continuity And Differentiability
Let \(f(x) = x.\left[ {{x \over 2}} \right]\), for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to _____.
INTEGER+4 / -02020
43Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2} + {x^4}} - 1}}\)
MCQ+4 / -12020
44Limits Continuity And Differentiability
Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y.
\(\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1\), then f'(3) is equal to ______.
\(\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1\), then f'(3) is equal to ______.
INTEGER+4 / -02020
45Limits Continuity And Differentiability
The function \(f(x) = \left\{ {\matrix{
{{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr
{{1 \over 2}\left( {\left| x \right| - 1} \right),} & {\left| x \right| > 1} \cr
} } \right.\) is :
MCQ+4 / -12020
46Limits Continuity And Differentiability
Let \(f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)\) be a differentiable function such that f(1) = e and \(\mathop {\lim }\limits_{t \to x} {{{t^2}{f^2}(x) - {x^2}{f^2}(t)} \over {t - x}} = 0\). If f(x) = 1, then x is equal ...
MCQ+4 / -12020
47Limits Continuity And Differentiability
Let [t] denote the greatest integer
\(\le\) t. If for some
\(\lambda\) \(\in\) R - {1, 0}, \(\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda - x + \left[ x \right]}}} \right|\) = L, then L is
equal ...
\(\le\) t. If for some
\(\lambda\) \(\in\) R - {1, 0}, \(\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda - x + \left[ x \right]}}} \right|\) = L, then L is
equal ...
MCQ+4 / -12020
48Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 0} \left\{ {{1 \over {{x^8}}}\left( {1 - \cos {{{x^2}} \over 2} - \cos {{{x^2}} \over 4} + \cos {{{x^2}} \over 2}\cos {{{x^2}} \over 4}} \right)} \right\}\) = 2-k
then the value of k is _______ .
then the value of k is _______ .
INTEGER+4 / -02020
49Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}\) (\(a\) \(\ne\) 0) is equa...
MCQ+4 / -12020
50Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}\) = 820,
(n \(\in\) N) then
the value of n is equal to _______.
(n \(\in\) N) then
the value of n is equal to _______.
INTEGER+4 / -02020