Limits, Continuity and Differentiability · Mathematics · MHT CET
Start PracticeMCQ (Single Correct Answer)
MHT CET 2023 14th May Evening Shift
The function $\mathrm{f}$ defined on $$\left(-\frac{1}{3}, \frac{1}{3}\right)$$ by
$$\mathrm{f}(x)=\left\{\begin{array}{cc}
\frac{1}{x} \log \left(\fr...
MHT CET 2023 14th May Evening Shift
If $$f(a)=2, f^{\prime}(a)=1, g(a)=-1, g^{\prime}(a)=2$$, then as $$x$$ approaches a, $$\frac{\mathrm{g}(x) \mathrm{f}(\mathrm{a})-\mathrm{g}(\mathrm{...
MHT CET 2023 14th May Evening Shift
Let $$\mathrm{f}(x)=5-|x-2|$$ and $$\mathrm{g}(x)=|x+1|, x \in \mathrm{R}$$ If $$\mathrm{f}(x)$$ attains maximum value at $$\alpha$$ and $$\mathrm{g}(...
MHT CET 2023 14th May Morning Shift
$$\lim _\limits{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=$$
MHT CET 2023 13th May Evening Shift
$$\lim _\limits{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=$$
MHT CET 2023 13th May Evening Shift
If $$f(x)$$ is continuous on its domain $$[-2,2]$$, where
$$f(x)=\left\{\begin{array}{cc}
\frac{\sin a x}{x}+3 & , \text { for }-2 \leq x...
MHT CET 2023 13th May Morning Shift
The function $$\mathrm{f}(\mathrm{t})=\frac{1}{\mathrm{t}^2+\mathrm{t}-2}$$ where $$\mathrm{t}=\frac{1}{x-1}$$ is discontinuous at
MHT CET 2023 13th May Morning Shift
If $$\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7$$, then $$\lim _\limits{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}=$$...
MHT CET 2023 12th May Evening Shift
$$\lim _\limits{x \rightarrow 0} \frac{x \cot 4 x}{\sin ^2 x \cdot \cot ^2(2 x)} \text { is equal to }$$
MHT CET 2023 12th May Evening Shift
Given $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x0\end{array}\right.$$
If $$\mathrm{f}(x)$$ is continuous at $$x...
MHT CET 2023 12th May Morning Shift
The values of $$a$$ and $$b$$, so that the function
$$f(x)=\left\{\begin{array}{l}
x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\
2 x \cot x+b, \...
MHT CET 2023 12th May Morning Shift
$$\lim _\limits{x \rightarrow 0} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2}$$ is
MHT CET 2023 11th May Evening Shift
$$\text { If } l=\lim _\limits{x \rightarrow 0} \frac{x}{|x|+x^2} \text {, then the value of } l \text { is }$$
MHT CET 2023 11th May Evening Shift
If $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{x-3}{|x-3|}+\mathrm{a} & , \quad x 3\end{array}\right.$$
Is continuous at $$x=3$$, then the value of ...
MHT CET 2023 11th May Morning Shift
$$\lim _\limits{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2}{x^3-3 x^2+2 x}\right]$$ is equal to
MHT CET 2023 11th May Morning Shift
The left-hand derivative of $$\mathrm{f}(x)=[x] \sin (\pi x)$$, at $$x=\mathrm{k}, \mathrm{k}$$ is an integer and [.] is the greatest integer function...
MHT CET 2023 11th May Morning Shift
If $$\mathrm{f}(x)=\left\{\begin{array}{ll}\frac{\sqrt{1+\mathrm{m} x}-\sqrt{1-\mathrm{m} x}}{x}, & -1 \leq x ...
MHT CET 2023 10th May Evening Shift
Let $$\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R} / \mathrm{f}(x)=|x-\pi|\left(\mathrm{e}^{|x|}-1\right) \sin |x|\right.$$ is not differentiable at $$...
MHT CET 2023 10th May Evening Shift
If $$\mathrm{f}(x)=\frac{4}{x^4}\left[1-\cos \frac{x}{2}-\cos \frac{x}{4}+\cos \frac{x}{2} \cdot \cos \frac{x}{4}\right]$$ is continuous at $$x=0$$, t...
MHT CET 2023 10th May Evening Shift
$$\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}=$$
MHT CET 2023 10th May Morning Shift
The value of $$\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}$$ is
MHT CET 2023 10th May Morning Shift
If the function $$\mathrm{f}(x)$$ is continuous in $$0 \leq x \leq \pi$$, then the value of $$2 a+3 b$$ is where
$$f(x)= \begin{cases}x+a \sqrt{2} \si...
MHT CET 2023 9th May Evening Shift
$$\lim _\limits{x \rightarrow 0} \frac{(1-\cos 2 x) \cdot \sin 5 x}{x^2 \sin 3 x}$$ is
MHT CET 2023 9th May Evening Shift
$$f(x)=\left\{\begin{array}{ll}
\frac{1-\cos k x}{x^2}, & \text { if } x \leq 0 \\
\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text { if } x>0
\end{arra...
MHT CET 2023 9th May Morning Shift
$$\lim _\limits{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^3}$$ equals
MHT CET 2022 11th August Evening Shift
$$\matrix{
{f(x) = a{x^2} + bx + 1,} & {if} & {\left| {2x - 3} \right| \ge 2} \cr
{ = 3x + 2,} & {if} & {{1 \over 2}
is continuous on its dom...
MHT CET 2022 11th August Evening Shift
If $$\lim _\limits{x \rightarrow 1} \frac{x^2-a x+b}{(x-1)}=5$$, then $$(a+b)$$ is equal to
MHT CET 2021 24th September Evening Shift
$$\lim _\limits{x \rightarrow 0} \frac{\sqrt{1-\cos x^2}}{1-\cos x}=$$
MHT CET 2021 24th September Evening Shift
If $$\mathrm{f}(\mathrm{x})=\mathrm{x}, \quad$$ for $$\mathrm{x} \leq 0$$
$$=0,\quad$$ for $$x>0$$, then the function $$f(x)$$ at $$x=0$$ is...
MHT CET 2021 24th September Morning Shift
$$\lim _\limits{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1}=$$
MHT CET 2021 24th September Morning Shift
If the function
$$\begin{array}{rlrl}
f(x) & =3 a x+b, & & \text { for } x1
\end{array}$$
is continuous at $$x=1$$. Then, the values of $$a$$ and $$b$...
MHT CET 2021 23rd September Evening Shift
$$\begin{aligned}
& \text { If the function given by} \mathrm{f}(\mathrm{x}) \\
& =-2 \sin \mathrm{x} \quad-\pi \leq \mathrm{x}
is continuous in $$[-\...
MHT CET 2021 23th September Morning Shift
If $$f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$$, for $$x \neq \pi$$ is continuous at $$x=\pi$$, then the value of $$f(\pi)$$ is
MHT CET 2021 23th September Morning Shift
$$\lim _\limits{x \rightarrow 1}\left[\frac{\sqrt{x}-1}{\log x}\right]=$$
MHT CET 2021 22th September Evening Shift
Let
$$\begin{aligned}
f(x) & =x+a \sqrt{2} \sin x & & , 0 \leq x
If $$\mathrm{f}(\mathrm{x})$$ is continuous for $$0 \leq \mathrm{x} \leq \pi$$, then...
MHT CET 2021 22th September Evening Shift
$$\lim _\limits{x \rightarrow 2}(x-1)^{ \frac{1}{3 x-6}}=$$
MHT CET 2021 22th September Morning Shift
Let
$$f(x)\matrix{
{ = |x| + 3,} & {if\,x \le - 3} \cr
{ = - 2x,} & {if\, - 3 ...
MHT CET 2021 22th September Morning Shift
$$\lim _\limits{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2}=$$
MHT CET 2021 21th September Evening Shift
$$\begin{aligned}
& \text { } f(x)=\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} \text {, if } 1 \leq x
is continuous in the interval $$[-1,1]$$, then $$p=$$
...
MHT CET 2021 21th September Evening Shift
If $$\lim _\limits{x \rightarrow 5} \frac{x^k-5^k}{x-5}=500$$, then the value of $$k$$, where $$k \in N$$ is
MHT CET 2021 21th September Morning Shift
$$\begin{aligned}
& \text { If the function } \mathrm{f}(\mathrm{x})=1+\sin \frac{\pi}{2}, \quad-\infty
is continuous in $$(-\infty, 6)$$, then the va...
MHT CET 2021 21th September Morning Shift
$$\lim _\limits{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=$$
MHT CET 2021 20th September Evening Shift
If $$a=\lim _\limits{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^2}$$ and $$b=\lim _\limits{n \rightarrow \infty} \frac{1^2+2^2+3^2+\ldots+n^2}{n^3}...
MHT CET 2021 20th September Evening Shift
If $$f(x) = {{{4^{x - \pi }} + {4^{x - \pi }} - 2} \over {{{(x - \pi )}^2}}}$$, for $$x \ne \pi $$, is continuous at $$x=\pi$$, then k =
MHT CET 2021 20th September Morning Shift
$$\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} + 5x - 7} - x} \right) = $$
MHT CET 2021 20th September Morning Shift
If f(x) = |x|, for x $$\in$$ ($$-1,2$$), then f is discontinuous at (where [x] represents floor function)
MHT CET 2020 19th October Evening Shift
If $f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$, for $x \neq \pi$ is continuous at $x=\pi$, then $f(\pi)=$
MHT CET 2020 16th October Evening Shift
If $$f(x)=\frac{|x|}{x}$$, for $$x \neq 0$$
$$=1$$, for $$x=0$$, then tre function is
MHT CET 2020 16th October Evening Shift
The function $$f(x)=\frac{x+1}{9 x+x^3}$$ is
MHT CET 2020 16th October Morning Shift
The points of discontinuity of the function
$$\begin{aligned}
f(x) & =\frac{1}{x-1}, \text { if } 0 \leq x \leq 2 \\
& =\frac{x+5}{x+3} \text { if } 2...
MHT CET 2019 2nd May Morning Shift
Which of the following function is not continuous at $x=0$ ?
MHT CET 2019 2nd May Morning Shift
If the function $f(x)=\frac{\log (1+a x)-\log (1-b x)}{x}$ $x \neq 0$ is continuous at $x=0$ then, $f(0)=\ldots \ldots$