Limits, Continuity and Differentiability PYQs - Last 10 Years

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Practice 159 MHT CET 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

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1Limits Continuity And Differentiability

Let $[x]$ denotes the greatest integer less than or equal to x and $f(x) = [\tan^2 x]$, then which of the following is true ?

2Limits Continuity And Differentiability

$\lim\limits_{x \to 0}\left[\dfrac{x \cdot \log(1 + 4x)}{\left(e^{4x} - 1\right)^2}\right] = \cdots$

3Limits Continuity And Differentiability

If the function $f(x)$ defined by $f(x) = \begin{cases} ax + 1 & \text{if } x \leq 3 \\ bx + 3 & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$, then $(a - b) =$ ..........

4Limits Continuity And Differentiability

The value of $\lim\limits_{n \to \infty}\left[\dfrac{1}{1 - n^2} + \dfrac{2}{1 - n^2} + \ldots + \dfrac{n}{1 - n^2}\right]^3$ is

5Limits Continuity And Differentiability

The value of $\lim\limits_{x \to 0}\left(\dfrac{8}{x^8}\right)\left[1 - \cos\dfrac{x^2}{2} - \cos\dfrac{x^2}{4} + \cos\dfrac{x^2}{2}\cdot\cos\dfrac{x^2}{4}\right]$ is equal to ...

6Limits Continuity And Differentiability

If the function $f(x) = \dfrac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x}$ is continuous at $x = \dfrac{\pi}{4}$, then the value of $f\left(\dfrac{\pi}{4}\right)$ is ...

7Limits Continuity And Differentiability

If the function $f(x) = \left(\dfrac{5x - 8}{8 - 3x}\right)^{\frac{3}{2x - 4}}$, for $x \neq 2$ is continuous at $x = 2$, then the value of $f(2)$ is...

8Limits Continuity And Differentiability

If $\lim_{x \to 1} \dfrac{x^3 + ax^2 + bx + c}{x^2 - 2x + 1} = 2026$ then the value of $a - c$ is...

9Limits Continuity And Differentiability

The number of point / points where the function $f(x) = \dfrac{1}{x^2-5|x|+6}$ is discontinuous is......

10Limits Continuity And Differentiability

If $\lim\limits_{x \to \infty}\left[\dfrac{x^2+x+1}{x+1} - ax - b\right] = 3$, then $a - b = $...

11Limits Continuity And Differentiability

If $f(x) = \dfrac{3^x + 3^{-x} - 2}{\tan x \cdot \log(1+x)}$ for $x \neq 0$, is continuous at $x = 0$, then the value of $f(0)$ is equal to $\ldots$

12Limits Continuity And Differentiability

If $\lim\limits_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2} = \lim\limits_{x \to 0} \dfrac{1 - \cos(2x)}{x \sin x}$, then the value of $k$ is $\ldots$

13Limits Continuity And Differentiability

Which of the following function is discontinuous at $x = 0$ ?

14Limits Continuity And Differentiability

If $\lim\limits_{x \to 1} \dfrac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b} = -2$, then the quadratic equation having roots $a$ and $b$ is

15Limits Continuity And Differentiability

If the function $f(x) = \dfrac{4\sqrt{2}(\sin 3x + \sin x)}{2\sin 2x\sin\dfrac{3x}{2} + \cos\dfrac{5x}{2} - \cos\dfrac{3x}{2}}$ for $x \neq \dfrac{\pi}{2}$ is continuous at $x = \dfrac{\pi}{2}$, then the value of $f\left(\dfrac{\pi}{2}\righ...

16Limits Continuity And Differentiability

If the derivative of the function $f(x) = \begin{cases} ax^2 + b & \text{if } x < -1 \\ bx^2 + ax + 4 & \text{if } x \geq -1 \end{cases}$ is continuous everywhere then

17Limits Continuity And Differentiability

$\displaystyle\lim_{x \to 0}\dfrac{(5^x - 1)^4\,\text{cosec}\,(x\log 5)}{\tan(x\log 5) \cdot \log(1 + x^2\log 25)} = \ldots\ldots$

18Limits Continuity And Differentiability

If the function f is continuous at $x = \pi$, where $f(x) = \dfrac{1 - \cos[7(x - \pi)]}{5(x - \pi)^2}$, for $x \neq \pi$, then $f(\pi) = $

19Limits Continuity And Differentiability

If $\lim\limits_{x \to 0}\dfrac{45^x - 9^x - 5^x + 1}{(k^x - 1)(3^x - 1)} = 2$, then the value of $k$ is ...

20Limits Continuity And Differentiability

If the function f is continuous at $x = 1$, where $f(x) = \dfrac{1 + \cos(\pi x)}{\pi(1-x)^2}$, for $x \neq 1$, then the value of $f(1)$ is....

21Limits Continuity And Differentiability

If $\lim\limits_{x \to 0} \dfrac{(4^x - 1)^3}{\tan\left(\dfrac{x}{4}\right)\log\left(1 + \dfrac{x^2}{3}\right)} = 96(\log a)^b$, then $(a + b) = $

22Limits Continuity And Differentiability

The value of f(0) so that the function $f(x) = \dfrac{(256 - 8x)^{\frac{1}{4}} - 4}{16 - 4(64 + 3x)^{\frac{1}{3}}}$, $x \neq 0$ is continuous at $x = 0$, is

23Limits Continuity And Differentiability

If $\lim\limits_{x \to \infty} \dfrac{(2x-1)^{19} \cdot (3x+2)^{11}}{(6x-5)^{30}} = 2^a \cdot 3^b$, then $a + b =$

24Limits Continuity And Differentiability

If $f(x) = \begin{cases} \dfrac{(8 - 2x)^{\frac{1}{3}} - 2}{3 - (243 + 5x)^{\frac{1}{5}}}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$, then $k =$

25Limits Continuity And Differentiability

$\displaystyle\lim_{x \to 0}\left[\dfrac{\log|2 + x| - \log|2 - x|}{\tan x}\right] = $ .......

26Limits Continuity And Differentiability

If $f(x)$ is continuous at $x = 0$, where $f(x) = \dfrac{8^x - 2^x}{k^x - 1}$, for $x \neq 0$ and $f(0) = 2$, then the value of $k$ is ...

27Limits Continuity And Differentiability

The value of $\lim_{n\to\infty} \dfrac{(n+2)! + (n+1)!}{(n+2)! - (n+1)!}$ is ____

28Limits Continuity And Differentiability

The quadratic polynomial $p(x)$ has roots $1$ and $\alpha$, while quadratic polynomial $q(x)$ has roots $1$ and $\beta$. Let $\alpha$ and $\beta$ be the roots of $r(x) = p(x) + q(x)$. Then $\lim_{x\to\infty}\left[\sqrt{p(x)} - \sqrt{q(x)}\r...

29Limits Continuity And Differentiability

Let the function $f(x)$ be defined as:$f(x) = \begin{cases} \left[\tan\left(\dfrac{\pi}{4} + x\right)\right]^{\dfrac{1}{x}}, & x \neq 0 \\ k, & x = 0 \end{cases}$If $f(x)$ is continuous at $x = 0$, then the value of $k$ is...

30Limits Continuity And Differentiability

The value of $\lim_{x \to 3} \dfrac{[x] - 3}{x - 3}$, where $[\cdot]$ denotes the greatest integer function, is.....

31Limits Continuity And Differentiability

Let the function f be defined by $f(x) = \dfrac{x - |x|}{x}$, for $x \neq 0$ and $f(0) = 2$ then f is

32Limits Continuity And Differentiability

$\lim_{x \to 1} \left[\dfrac{x-2}{x^2-x} - \dfrac{1}{x^3 - 3x^2 + 2x}\right] = $

33Limits Continuity And Differentiability

Define $f(x)=\left\{\begin{array}{cl}b-a x & , \text { if } x<2 \\ 3 & , \text { if } x=2 \\ a+2 b x & , \text { if } x>2\end{array}\right.$ and if $\lim _{x \rightarrow 2} f(x)$ exists, then $\frac{a}{b}=$

34Limits Continuity And Differentiability

If $\quad f(x)=\left\{\begin{array}{cc}\frac{9^x-2 \cdot 3^x+1}{\log (1+3 x) \cdot \tan 2 x} & , \text { if } x \neq 0 \\ a(\log b)^c & , \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}=$

35Limits Continuity And Differentiability

If the function $f(x)=\left\{\begin{array}{cl}\frac{\cos a x-\cos b x}{\cos c x-\cos b x} & , \text { if } x \neq 0 \\ -1 & , \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $\mathrm{a}^2, \mathrm{~b}^2, \mathrm{c}^2$ are i...

36Limits Continuity And Differentiability

\(\mathop {\lim }\limits_{x \to \infty } \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}=\)

37Limits Continuity And Differentiability

\(\mathop {\lim }\limits_{x \to 0} \frac{\mathrm{e}^{x^2}-\cos 3 x}{\sin x \log (1+2 x)}=\)

38Limits Continuity And Differentiability

Let $f(x)= \begin{cases}\frac{x^4-5 x^2+4}{|(x-1)(x-2)|} & , x \neq 1,2 \\ 6 & , x=1 \\ 12 & , x=2\end{cases}$
Then $\mathrm{f}(x)$ is continuous on the set

39Limits Continuity And Differentiability

If $\mathrm{f}(x)=\frac{\sin \left(\pi \cos ^2 x\right)}{3 x^2}, x \neq 0$ is continuous at $x=0$ then $\mathrm{f}(0)=$

40Limits Continuity And Differentiability

If Rolle's theorem holds for the function $x^3+\mathrm{a} x^2+\mathrm{b} x, 1 \leq x \leq 2$ at the point $\frac{4}{3}$, then the values of $a$ and $b$ are respectively

41Limits Continuity And Differentiability

\(\lim\limits_{x \rightarrow 5} \frac{\sqrt{2-2 \cos \left(x^2-12 x+35\right)}}{(x-5)}=\ldots \ldots\)

42Limits Continuity And Differentiability

\(\mathop {\lim }\limits_{x \to 0} \frac{63^x-9^x-7^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=\ldots \ldots\)

43Limits Continuity And Differentiability

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable function having $\mathrm{f}(3)=3, \mathrm{f}^{\prime}(3)=\frac{1}{27}$ and $\mathrm{g}(x)= \begin{cases}\int_3^{\mathrm{f}(x)} \frac{3 \mathrm{t}^2}{x-3} \mathrm{dt}, & \...

44Limits Continuity And Differentiability

If $\mathrm{f}(x)=\frac{10^x+7^x-14^x-5^x}{1-\cos x}, x \neq 0$ is continuous at $x=0$, then the value of $\mathrm{f}(0)$ is

45Limits Continuity And Differentiability

\(\mathop {\lim }\limits_{x \to 2} \frac{x+3 x^2+5 x^3+7 x^4-166}{x-2}=\)

46Limits Continuity And Differentiability

Let $\mathrm{A}=\mathop {\lim }\limits_{x \to {0^ + }}\left(1+\tan ^2 \sqrt{x}\right)^{\frac{1}{2 x}}$, then $\log _{\mathrm{e}} \mathrm{A}=$

47Limits Continuity And Differentiability

The function $\mathrm{f}(x)=2 x-\left|x-x^2\right|$ is

48Limits Continuity And Differentiability

\(\lim\limits_{x \rightarrow \infty}\left(\frac{x+8}{x+1}\right)^{x+5}=\ldots\)

49Limits Continuity And Differentiability

If the function
$$ f(x)=\left\{\begin{array}{cc} x+a \sqrt{2} \sin x & \text { if } 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b & \text { if } \frac{\pi}{4} < x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x & \text { if } \frac{\pi}{2} < x \leq...

50Limits Continuity And Differentiability

If $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x<0 \\ \frac{a}{\sqrt{x}} & , \text { if } x=0 \\ \frac{(16+\sqrt{x})^{\frac{1}{2}}-4}{16} & , \text { if } x>0\end{array}\right.$
is continuous at $x=0$, then $\mathr...