Limits, Continuity and Differentiability PYQs - Last 5 Years
JEE Main / Mathematics / Calculus / 128 recent questions
MathematicsCalculus2022-2026
Practice 128 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.
128
PYQs on Page
Mathematics / Calculus
2022-2026
Year Range
Based on indexed question metadata
128
Last 5 Years
2022-2026
128
Last 10 Years
2017-2026
Recent Year Trend
2022
2023
2024
2025
2026Latest year
202233 max PYQs/year2026
Question Types
128PYQs
MCQ71.1%
INTEGER28.9%
Difficulty Mix
#1 Medium95
#2 Hard32
#3 Easy1
128 in last 5 years128 in last 10 years
Last 5 Years Limits, Continuity and Differentiability Questions
Showing 28 of 128 filtered questions.
1Limits Continuity And Differentiability
$$
\text { Let the function } f(x)=\left\{\begin{array}{cl}
\frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\
10 & ; \text { if } x=0
\end{array} \text { be continuous at } x=0 .\right.
$$
Then \(\alpha\) is equal...
Then \(\alpha\) is equal...
MCQ+4 / -12022
2Limits Continuity And Differentiability
Let f : R \(\to\) R be defined as
\(f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.\)
w...
\(f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.\)
w...
MCQ+4 / -12022
3Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2\), then the value of (a \(-\) b) is equal to ___________.
INTEGER+4 / -12022
4Limits Continuity And Differentiability
The value of \(\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}\) is equal to :
MCQ+4 / -12022
5Limits Continuity And Differentiability
Let f, g : R \(\to\) R be functions defined by
\(f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{e^x} - x} & , & {x < 0} \cr
{{{(x - 1)}^2} - ...
\(f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{e^x} - x} & , & {x < 0} \cr
{{{(x - 1)}^2} - ...
MCQ+4 / -12022
6Limits Continuity And Differentiability
\(\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}\) is equal to ___________.
INTEGER+4 / -12022
7Limits Continuity And Differentiability
Let \(f:[0,1] \rightarrow \mathbf{R}\) be a twice differentiable function in \((0,1)\) such that \(f(0)=3\) and \(f(1)=5\). If the line \(y=2 x+3\) intersects the graph of \(f\) at only two distinct points in \((0,1)\), then the least numbe...
INTEGER+4 / -12022
8Limits Continuity And Differentiability
The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\lim\limits_{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is continuous for all x in :
MCQ+4 / -12022
9Limits Continuity And Differentiability
Let a be an integer such that \(\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}\) exists, where [t] is greatest integer \(\le\) t. Then a is equal to :
MCQ+4 / -12022
10Limits Continuity And Differentiability
Let [t] denote the greatest integer \(\le\) t and {t} denote the fractional part of t. The integral value of \(\alpha\) for which the left hand limit of the function
$$f(x) = [1 + x] + {{{\alpha ^{2[x] + {\{x\}}}} + [x] - 1} \over {2[x] + \...
$$f(x) = [1 + x] + {{{\alpha ^{2[x] + {\{x\}}}} + [x] - 1} \over {2[x] + \...
INTEGER+4 / -12022
11Limits Continuity And Differentiability
If for \(\mathrm{p} \neq \mathrm{q} \neq 0\), the function \(f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}\) is continuous at \(x=0\), then :
MCQ+4 / -12022
12Limits Continuity And Differentiability
Let f, g : R \(\to\) R be two real valued functions defined as \(f(x) = \left\{ {\matrix{
{ - |x + 3|} & , & {x < 0} \cr
{{e^x}} & , & {x \ge 0} \cr
} } \right.\) and $$g(x) = \left\{ {\matrix{
{{x^2} + {k_1}x} & , & {x < 0} ...
{{x^2} + {k_1}x} & , & {x < 0} ...
MCQ+4 / -12022
13Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}\) is equal to :
MCQ+4 / -12022
14Limits Continuity And Differentiability
Let f(x) = min {1, 1 + x sin x}, 0 \(\le\) x \(\le\) 2\(\pi\). If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to
MCQ+4 / -12022
15Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}\) is equal to :
MCQ+4 / -12022
16Limits Continuity And Differentiability
Let \(f(x) = \left\{ {\matrix{
{{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr
{ - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr
} } \right.\).
Then the set of all values of b, for which f(x) has maximum value at x = 1, is :
Then the set of all values of b, for which f(x) has maximum value at x = 1, is :
MCQ+4 / -12022
17Limits Continuity And Differentiability
If \(f(x) = \left\{ {\matrix{
{x + a} & , & {x \le 0} \cr
{|x - 4|} & , & {x > 0} \cr
} } \right.\) and \(g(x) = \left\{ {\matrix{
{x + 1} & , & {x < 0} \cr
{{{(x - 4)}^2} + b} & , & {x \ge 0} \cr
} } \right.\) are c...
MCQ+4 / -12022
18Limits Continuity And Differentiability
If the function $$f(x) = \left\{ {\matrix{
{{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr
k & , & {x = 0} \cr
...
{{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr
k & , & {x = 0} \cr
...
MCQ+4 / -12022
19Limits Continuity And Differentiability
Let f : R \(\to\) R be a continuous function such that \(f(3x) - f(x) = x\). If \(f(8) = 7\), then \(f(14)\) is equal to :
MCQ+4 / -12022
20Limits Continuity And Differentiability
Let \(\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}\) for some \(\alpha \in \mathbb{R}\). Then the value of \(\alpha+\beta\) is :
MCQ+4 / -12022
21Limits Continuity And Differentiability
Let f(x) be a polynomial function such that \(f(x) + f'(x) + f''(x) = {x^5} + 64\). Then, the value of \(\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}\) is equal to:
MCQ+4 / -12022
22Limits Continuity And Differentiability
Let \(f(x) = \left[ {2{x^2} + 1} \right]\) and \(g(x) = \left\{ {\matrix{
{2x - 3,} & {x < 0} \cr
{2x + 3,} & {x \ge 0} \cr
} } \right.\), where [t] is the greatest integer \(\le\) t. Then, in the open interval (\(-\)1, 1), the ...
INTEGER+4 / -12022
23Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)\) is equal to
MCQ+4 / -12022
24Limits Continuity And Differentiability
Let $$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^{2}-6 x+1<0,}\end{array}\right.$$ where \([\alpha]\) denotes the greatest integer less tha...
INTEGER+4 / -12022
25Limits Continuity And Differentiability
If \(\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0\), then \(8(\alpha+\beta)\) is equal to :
MCQ+4 / -12022
26Limits Continuity And Differentiability
\(\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}\) is equal to
MCQ+4 / -12022
27Limits Continuity And Differentiability
The number of points where the function
\(f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1 < x < 1} \cr {|x + 1| + |x - 2|} & {if} & {x \ge 1} \cr } } \right.\)
[t] denote...
\(f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1 < x < 1} \cr {|x + 1| + |x - 2|} & {if} & {x \ge 1} \cr } } \right.\)
[t] denote...
INTEGER+4 / -12022
28Limits Continuity And Differentiability
Let \(f(x) = \left\{ {\matrix{
{{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr
{\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr
1 & {,\,otherwise} \cr
} } \right.\)
where [t] denotes greatest integer \(\le\) t. I...
where [t] denotes greatest integer \(\le\) t. I...
MCQ+4 / -12022