Limits, Continuity and Differentiability PYQs - Last 10 Years
TS EAMCET / Mathematics / Calculus / 74 recent questions
MathematicsCalculus2016-2025
Practice 74 TS EAMCET 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 74 filtered questions.
1Limits Continuity And Differentiability
If $\mathop {\lim }\limits_{x \to 0} \frac{3^{x^3}-\left(1-x^3\right)^{\frac{2}{3}}}{x^2 \sin x}=p+\log q$, then $p q=$
MCQ+1 / -02025
2Limits Continuity And Differentiability
If $[x]$ is the greatest integer function and
$$ f(x)=\left\{\begin{array}{cc} 2[x]-\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{array}\right. $$
is a real valued function, then $f$ is
$$ f(x)=\left\{\begin{array}{cc} 2[x]-\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{array}\right. $$
is a real valued function, then $f$ is
MCQ+1 / -02025
3Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin ^2 x}=\)
MCQ+1 / -02025
4Limits Continuity And Differentiability
The set of all values of $x$ for which $f(x)=\| x|-1|$ is differentiable is
MCQ+1 / -02025
5Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cc}x^2\left|\cos \frac{\pi}{2}\right|, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at $x=2, f(x)$ is
MCQ+1 / -02025
6Limits Continuity And Differentiability
Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=\left[x^2-3\right]$ where $[$. denotes greatest integer function, then the number of points of discontinuity for the function $f$ in $(-1,2)$ is
MCQ+1 / -02025
7Limits Continuity And Differentiability
The value of $x$ at which the real valued function $f(x)=7|2 x+1|-19|3 x-5|$ is not differentiable is
MCQ+1 / -02025
8Limits Continuity And Differentiability
For $a \neq 0$ and $b \neq 0$, if the real valued function $f(x)=\frac{\sqrt[5]{a(625+x)}-5}{\sqrt[4]{625+b x}-5}$ is continuous at $x=0$, then $f(0)=$
MCQ+1 / -02025
9Limits Continuity And Differentiability
If $\{x\}=x-[x]$, where $[x]$ is the greatest integer $\leq x$ and $\mathop {\lim }\limits_{x \to {0^ - }} \frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^4}=\theta$, then $\tan \theta$
MCQ+1 / -02025
10Limits Continuity And Differentiability
If $[t]$ represents the greatest integer $\leq t$, then the value of $\lim\limits_{x \rightarrow 3} \frac{11-[2-x]}{[x+10]}$ is
MCQ+1 / -02025
11Limits Continuity And Differentiability
If the real valued function
$$ f(x)=\left\{\begin{array}{ccc} \frac{\cos 3 x-\cos x}{x \sin x}, & \text { if } & x<0 \\ p, & \text { if } & x=0 \\ \frac{\log (1+q \sin x)}{x}, & \text { if } & x>0 \end{array}\right. $$
is continuous at $x=0...
$$ f(x)=\left\{\begin{array}{ccc} \frac{\cos 3 x-\cos x}{x \sin x}, & \text { if } & x<0 \\ p, & \text { if } & x=0 \\ \frac{\log (1+q \sin x)}{x}, & \text { if } & x>0 \end{array}\right. $$
is continuous at $x=0...
MCQ+1 / -02025
12Limits Continuity And Differentiability
Let ' $a$ ' be a positive real number. If a real valued function
$f(x)=\left\{\begin{array}{cl}\frac{6^x-3^x-2^x+1}{1-\cos \left(\frac{x}{a}\right)} & \text { if } x \neq 0 \\ \log 3 \log 4 & \text { if } x=0\end{array}\right.$ is continuou...
$f(x)=\left\{\begin{array}{cl}\frac{6^x-3^x-2^x+1}{1-\cos \left(\frac{x}{a}\right)} & \text { if } x \neq 0 \\ \log 3 \log 4 & \text { if } x=0\end{array}\right.$ is continuou...
MCQ+1 / -02025
13Limits Continuity And Differentiability
If $[x]$ is the greatest integer function, then
\(\mathop {\lim }\limits_{x \to 3} \frac{(3-|x|+\sin |3-x|) \cos [9-3 x]}{|3-x|[3 x-9]}\)
\(\mathop {\lim }\limits_{x \to 3} \frac{(3-|x|+\sin |3-x|) \cos [9-3 x]}{|3-x|[3 x-9]}\)
MCQ+1 / -02025
14Limits Continuity And Differentiability
If $f(x)=\left\{\begin{array}{cc}\frac{a \sin x-b x+c x^2+x^3}{2 \log (1+x)-2 x^3+x^4} & , x \neq 0 \\ 0 & , x=0\end{array}\right.$
is continuous at $x=0$, then
is continuous at $x=0$, then
MCQ+1 / -02025
15Limits Continuity And Differentiability
If the function $g(x)=\left\{\begin{array}{cl}K \sqrt{x+1} & , 0 \leq x \leq 3 \\ m x+2 & , 3 < x \leq 5\end{array}\right.$ is differentiable, then $K+m=$
MCQ+1 / -02025
16Limits Continuity And Differentiability
If $f(x)=\frac{x\left(a^x-1\right)}{1-\cos x}$ and $g(x)=\frac{x\left(1-a^x\right)}{a^x\left(\sqrt{1-x^2}-\sqrt{1+x^2}\right)}$, then $\lim _{x \rightarrow 0}(f(x)-g(x))=$
MCQ+1 / -02025
17Limits Continuity And Differentiability
If the function
$$ f(x)=\left\{\begin{array}{cc} \frac{\cos a x-\cos 9 x}{x^2} & \text {, if } x \neq 0 \\ 16 & \text {, if } x=0 \end{array}\right. $$
is continuous at $x=0$, then $a=$
$$ f(x)=\left\{\begin{array}{cc} \frac{\cos a x-\cos 9 x}{x^2} & \text {, if } x \neq 0 \\ 16 & \text {, if } x=0 \end{array}\right. $$
is continuous at $x=0$, then $a=$
MCQ+1 / -02024
18Limits Continuity And Differentiability
$\lim \limits_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots .(2)\right]^{1 / n}=$
MCQ+1 / -02024
19Limits Continuity And Differentiability
$\lim _{x \rightarrow 0} \frac{3^{\sin x}-2^{\tan x}}{\sin x}=$
MCQ+1 / -02024
20Limits Continuity And Differentiability
If $ f(x)=\left\{\begin{array}{ll}\frac{8}{x^{3}}-6 x & \text {, if } 0 < x \leq 1 \\\\ \frac{x-1}{\sqrt{x}-1} & \text {,if } x > 1\end{array}\right. $ is a real valued function, then at $ x=1, f $ is
MCQ+1 / -02024
21Limits Continuity And Differentiability
Define $ f: R \rightarrow R $ by $ f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^{2}}, & x < 0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0\end{array}\right. $
Then, the value of $ a $ so that $ f $ is continuous at $ x=...
Then, the value of $ a $ so that $ f $ is continuous at $ x=...
MCQ+1 / -02024
22Limits Continuity And Differentiability
$\lim _{\theta \rightarrow \frac{\pi^{-}}{2}} \frac{8 \tan ^4 \theta+4 \tan ^2 \theta+5}{(3-2 \tan \theta)^4}=$
MCQ+1 / -02024
23Limits Continuity And Differentiability
If the real valued function $f(x)=\int \frac{\left(4^{x}-1\right)^{4} \cot (x \log 4)}{\sin (x \log 4) \log \left(1+x^{2} \log 4\right)}, \quad$ if $x \neq 0$ is continuous at $x=0$, then $e^{k}=$
MCQ+1 / -02024
24Limits Continuity And Differentiability
$\lim\limits_{x \rightarrow \frac{3}{2}} \frac{\left(4 x^{2}-6 x\right)\left(4 x^{2}+6 x+9\right)}{\sqrt[3]{2 x}-\sqrt[3]{3}}=$
MCQ+1 / -02024
25Limits Continuity And Differentiability
If $\lim \limits_{x \rightarrow 4} \frac{2 x^2+(3+2 a) x+3 a}{x^3-2 x^2-23 x+60}=\frac{11}{9}$, then $\lim \limits_{x \rightarrow a} \frac{x^2+9 x+20}{x^2-x-20}=$
MCQ+1 / -02024
26Limits Continuity And Differentiability
If the function
$$ f(x)= \begin{cases}\frac{\tan a(x-1)}{x-1}, & \text { if } 04\end{cases} $$
domain, then $6 a+9 b^4=$
$$ f(x)= \begin{cases}\frac{\tan a(x-1)}{x-1}, & \text { if } 04\end{cases} $$
domain, then $6 a+9 b^4=$
MCQ+1 / -02024
27Limits Continuity And Differentiability
If the function $f(x)=\left\{\begin{array}{cl}\frac{\left(e^{k x}-1\right) \sin k x}{4 \tan x} & x \neq 0 \\ P & x=0\end{array}\right.$ is differentiable at $x=0$, then
MCQ+1 / -02024
28Limits Continuity And Differentiability
If Rolle's Theorem is applicable for the function $f(x)=\left\{\begin{array}{cl}x^{p} \log x, & x \neq 0 \\ 0, & x=0\end{array}\right.$ on the interval $[0,1]$, then a possible value of $p$ is
MCQ+1 / -02024
29Limits Continuity And Differentiability
If $0 \leq x \leq \frac{\pi}{2}$, then $\lim _{x \rightarrow a} \frac{|2 \cos x-1|}{2 \cos x-1}$
MCQ+1 / -02024
30Limits Continuity And Differentiability
If $f(x)=3 x^{15}-5 x^{10}+7 x^{5}+50 \cos (x-1)$, then $\lim\limits_{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}$
MCQ+1 / -02024
31Limits Continuity And Differentiability
The real valued function $f(x)=\frac{|x-a|}{x-a}$ is
MCQ+1 / -02024
32Limits Continuity And Differentiability
\(\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n\left(k^2 x\right)=\)
MCQ+1 / -02023
33Limits Continuity And Differentiability
The quadratic equation whose roots are
$$ l=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right) \text { and } m=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\r...
$$ l=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right) \text { and } m=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\r...
MCQ+1 / -02023
34Limits Continuity And Differentiability
$$ \begin{array}{r} \lim _{x \rightarrow 0} \frac{2 \tan x+\cos x-1+x}{\sqrt{4 \sin ^2 x+2 \tan x+1}}= \\ -\sqrt{3 \tan ^2 x+\sin x+1} \end{array} $$
MCQ+1 / -02023
35Limits Continuity And Differentiability
If a function $f$ is defined by $f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)},(x \neq \pi / 4)$, then $\lim _{x \rightarrow \pi / 4} f(x)=$
MCQ+1 / -02023
36Limits Continuity And Differentiability
\(\lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}=\)
MCQ+1 / -02023
37Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 2 + }\left([x]^2-[x]-2\right)+\mathop {\lim }\limits_{x \to- 3 - }\left([x]^2-4[x]+3\right)=\)
MCQ+1 / -02023
38Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 2} \frac{\sqrt[3]{6+x}-\sqrt[3]{10-x}}{x-2}=\)
MCQ+1 / -02023
39Limits Continuity And Differentiability
$\mathop {\lim }\limits_{x \to 0} \frac{\tan ^4 x-\sin ^4 x}{x^6}=$
MCQ+1 / -02023
40Limits Continuity And Differentiability
\(\lim \limits_{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=\)
MCQ+1 / -02023
41Limits Continuity And Differentiability
If $a, b, c$ and $k$ are non-zero real numbers and $\lim \limits_{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$, then $k=$
MCQ+1 / -02023
42Limits Continuity And Differentiability
\(\lim\limits_{x \rightarrow 1}(1-x) \tan \left(\frac{\pi}{2} x\right)=\)
MCQ+1 / -02023
43Limits Continuity And Differentiability
If $f(9)=9$ and $f^{\prime}(9)=4$, then $\lim\limits_{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=$
MCQ+1 / -02023
44Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \frac{\left(2^x-1\right)(1+\sin x)^{\frac{2}{\sin x}}}{\log (1+2 x)}=\)
MCQ+1 / -02022
45Limits Continuity And Differentiability
Let [ $x$ ] denote the greatest integer less than or equal to $x$ and $f(x)=2 x-[2 x]$. If $\mathop {\lim }\limits_{x \to {2^ - }} f(x)=l_1$ and $\mathop {\lim }\limits_{x \to {2^ + }} f(x)=l_2$, then $l_1+l_2=$
MCQ+1 / -02022
46Limits Continuity And Differentiability
Let $f(x)$ be a differentiable function such that $f(0)=0$ and $f^{\prime}(0)=20$. For $x \in\left(0, \frac{\pi}{2}\right]$, if
$A(x)=2 f(x) \operatorname{cosec} 4 x+4 f(x)\left(\cos ^2 x+1\right)-4 \cos ^2 x$, then $\mathop {\lim }\limits_...
$A(x)=2 f(x) \operatorname{cosec} 4 x+4 f(x)\left(\cos ^2 x+1\right)-4 \cos ^2 x$, then $\mathop {\lim }\limits_...
MCQ+1 / -02022
47Limits Continuity And Differentiability
\(\lim _{x \rightarrow 2} \frac{x^3-x^2-x-2}{2 x^3-3 x^2-3 x+2}=\)
MCQ+1 / -02022
48Limits Continuity And Differentiability
\(\lim _{x \rightarrow 0} \frac{4[\sin (2022 x)-\sin (2020 x)]}{x[\cos (2022 x)+2 \cos (2021 x)+\cos (2020 x)]}=\)
MCQ+1 / -02022
49Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0}\left(\frac{4!}{x^8}\left(1-\cos \frac{x^2}{3}-\cos \frac{x^2}{4}+\cos \frac{x^2}{3} \cos \frac{x^2}{4}\right)\right)=\)
MCQ+1 / -02022
50Limits Continuity And Differentiability
\(\mathop {\lim }\limits_{x \to 0} \frac{\tan ^2\left(\pi \sec ^4 x\right)}{\pi^2 x^4}=\)
MCQ+1 / -02022