Q JEE Main 2024

Let a>0 be a root of the equation $2 \mathrm{x}^2+\mathrm{x}-2=0$. If $\lim _{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{\left(1-a x^2\right)}=\alpha+\beta \sqrt{17}$,...

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Q JEE Main 2024

$\operatorname{Lim}_{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to :

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Q Jee Main 2024

If $^2$-blog $(1+x)+\quad \lim _{x \rightarrow 0}$ $\_\_\_\_$。 = 1 axes $\cos x$, then $16\left(a^2+b^2+c^2\right)$ is equal to $$ x^2...

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Q JEE MAIN 2024

Let $...

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Q JEE-Main 2024

If $\lim _{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3}$, then $2 \alpha-\beta$ is equal to...

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Q JEE MAIN 2026

The value of $...

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Q JEE MAIN 2026

The value of $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is

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Q JEE MAIN 2026

If $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to:

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Q JEE MAIN_2019

If $\lim _{x \rightarrow 1} \frac{x^4-1}{x-1}=\lim _{x \rightarrow *} \frac{x^3-k^3}{x^2-k^2}$, then $k$ is

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^2+2 \sin x+1}-\sqrt{\sin ^2 x-x+1}}$ is :

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Q JEE Main 2020

If $\lim _{x \rightarrow 1} \frac{x+x^2+x^3+\ldots .+x^n-n}{x-1}=820,(n \in N)$ then the value of $n$ is equal to $\_\_\_\_$

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}$ is equal to :

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Q JEE MAIN 2019

For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $...

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Q JEE MAIN 2020

If $\alpha$ is the positive root of the equation, $p(x)=x^2-x -2=0$, then $\lim _{x \rightarrow \alpha^{+}} \frac{\sqrt{1-\cos (p(x))}}{x+\alpha-4}$ is equal...

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Q JEE MAIN 2019

$\lim _{x \rightarrow 1} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}}$ is equal to :

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Q JEE MAIN 2019

For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $...

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Q JEE MAIN 2019

Let $[\mathrm{x}]$ denote the greatest integer less than or equal to x . Then $...

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Q JEE MAIN 2020

$\lim _{x \rightarrow 0}\left(\frac{3 x^2+2}{7 x^2+2}\right)^{\frac{1}{x^2}}$ is equal to

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Q JEE MAIN 2019

If $\lim _{x \rightarrow 1} \frac{x^2-a x+b}{x-1}=5$, then $a+b$ is equal to :

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Q JEE MAIN_2021_

If $\lim _{x \rightarrow \infty}\left(\sqrt{x^2-x+1}-a x\right)=b$, then the ordered pair $(a b)$ is :

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Q JEE MAIN_2021_

If $0

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Q JEE-MAIN 2020

$\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^2+x^4}-1\right) / x}-1\right)}{\sqrt{1+x^2+x^4}-1}$

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Q JEE Main 2019

For each tR, let [t] be the greatest integer less than or equal to t. Then,

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Q JEE-MAIN 2020

$\lim _{x \rightarrow a} \frac{(a+2 x)^{\frac{1}{3}}-(3 x)^{\frac{1}{3}}}{(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}}(a \neq 0)$ is equal to : $(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}$

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{x \cot (4 x)}{\sin ^2 x \cot ^2(2 x)}$ is equal to

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Q JEE MAIN 2020

$\lim _{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2019

If $f(x)=[x]-\left[\frac{x}{4}\right], x \in R$, where $[x]$ denotes the greatest integer function, then :

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Q JEE MAIN 2021

$\lim _{x \rightarrow 0} \frac{\sin ^2\left(\pi \cos ^4 x\right)}{x^4}$ is equal to:

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Q JEE Main 2021

If $\alpha=\lim _{x \rightarrow \pi / 4} \frac{\tan ^3 x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$ and $\beta=\lim _{x \rightarrow 0}(\cos x)^{\cot x}$ are...

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Q JEE MAIN 2021

If $\alpha, \beta$ are the distinct roots of $x^2+b x+c=0$, then $\lim _{x \rightarrow \beta} \frac{e^{2\left(x^2+b x+c\right)}-1-2\left(x^2+b x+c\right)}{(x-\beta)^2}$ is equal...

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Q JEE MAIN 2021

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Q JEE MAIN 2019

$\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals :

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Q JEE Main 2019

$$ \lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^2}}-\sqrt{2}}{y^4} $$

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Q JEE MAIN 2021

$\lim _{x \rightarrow 2}\left(\sum_{n=1}^9 \frac{x}{n(n+1) x^2+2(2 n+1) x+4}\right)$ is equal to :

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Q JEE MAIN 2021

Let $f(x)=x^6+2 x^4+x^3+2 x+3, x \in R$. Then the natural number $n$ for which $\lim _{x \rightarrow 1} \frac{x^n f(1)-f(x)}{x-1}=44$...

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Q JEE MAIN 2021 S2

The value of $\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to:

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Q JEE MAIN 2019

Let $f: R \rightarrow R$ be a differentiable function satifying $f^{\prime}(3)+f^{\prime}(2)=0$. Then $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}$ is equal to :

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Q JEE MAIN 2021

Let $f: R \rightarrow R$ be a function such that $f(2)=4$ and $f(2)=1$. Then, the value of $...

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Q JEE MAIN 2021

If the value of $\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then a is equal...

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{a x-\left(e^{4 x}-1\right)}{a x\left(e^{4 x}-1\right)}$ exists and is equal to $b$, then the value of...

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{\alpha x e^x-\beta \log _e(1+x)+\gamma x^2 e^{-x}}{x \sin ^2 x}=10, \alpha, \beta, \gamma \in R$,...

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Q JEE-MAIN 2021

The value of the limit $\lim _{\theta \rightarrow 0} \frac{\tan \left(\pi \cos ^2 \theta\right)}{\left(2 \pi \sin ^2 \theta\right)}$ equal to:

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Q JEE-MAIN 2021

The value of $\lim _{n \rightarrow \infty} \frac{[r]+[2 r]+\ldots \ldots+[n r]}{n^2}$ where $r$ is non-zero real number and $[r]$ denotes...

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Q JEE MAIN_2021_

The value of

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty} \tan \left\{\sum_{r=1}^n \tan ^{-1}\left(\frac{1}{1+r+r^2}\right)\right\}$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2021

If $\lim _{x \rightarrow 0} \frac{\sin ^{-1} x-\tan ^{-1} x}{3 x^2}$ is equal to $L$, then the value of $...

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Q JEE-MAIN 2021

If $\lim _{x \rightarrow 0} \frac{a e^x-b \cos x+c e^{-x}}{x \sin x}=2$, then $a+b+c$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2021

Let $S_k=\sum_{r=1}^k \tan ^{-1}\left(\frac{6^r}{2^{2 r+1}+3^{2 r+1}}\right)$. Then $\lim _{k \rightarrow \infty} S_k$ is equal to :

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Q JEE MAIN 2021

Let $f(x)=\sin ^{-1} x$ and $g(x)=\frac{x^2-x-2}{2 x^2-x-6}$. If $g(2)=\lim _{x \rightarrow 2} g(x)$, then the domain of the function fog...

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Q JEE MAIN 2021

The value of $\lim _{h \rightarrow 0} 2\left\{\frac{\sqrt{3} \sin \left(\frac{\pi}{6}+h\right)+\cos \left(\frac{\pi}{6}+h\right)}{\sqrt{3} h(\sqrt{3} \cos h-\sin h)}\right\}$ is

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Q JEE MAIN 2021

$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots+\frac{1}{n}}{n^2}\right)^n$ is equal to :

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Q JEE MAIN 2022

If $\lim _{n \rightarrow \infty}\left(\sqrt{n^2-n-1}+n \alpha+\beta\right)=0$ then $8(\alpha+\beta)$ is equal to :

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Q JEE MAIN 2022

Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. Then integral value of...

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Q JEE MAIN 2022

If $\lim _{x \rightarrow 1} \frac{\sin \left(3 x^2-4 x+1\right)-x^2+1}{2 x^3-7 x^2+a x+b}=-2$, then the value of $(a-b)$ is equal to

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Q JEE MAIN 2022

The value of $\lim _{x \rightarrow 1} \frac{\left(x^2-1\right) \sin ^2(\pi x)}{x^4-2 x^3+2 x-1}$ is equal to:

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Q JEE MAIN 2022

$\lim _{n \rightarrow \infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\ldots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$ is equal to

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Q JEE MAIN 2022

$\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^4}$ is equal to :

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Q JEE MAIN 2022

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\tan ^2 x\left(\left(2 \sin ^2 x+3 \sin x+4\right)^{\frac{1}{2}}-\left(\sin ^2 x+6 \sin x+2\right)^{\frac{1}{2}}\right)\right)$ is equal to

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Q JEE MAIN 2022

Let a be an integer such that $\lim _{x \rightarrow 7} \frac{18-[1-x]}{[x-3 a]}$ exists, where $[t]$ is greatest integer $...

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Q JEE MAIN 2022

$\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to :

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Q [JEE-Main 2023

If $\alpha>\beta>0$ are the roots of the equation $a x^2+b x+1=0$, and $\lim _{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^2+b x+a\right)}{2(1-\alpha x)^2}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right)$, then...

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Q JEE MAIN_2025

If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $x=0$, then $f(0)$ is equal to $\_\_\_\_$...

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Q JEE MAIN 2025

Given below are two statements: Statement I : $\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}$ Statement II: $...

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Q JEE MAIN_2025

For $\mathrm{t}>-1$, let $\alpha_{\mathrm{t}}$ and $\beta$, be the roots of the equation $$ \left((t+2)^{1 / 7}-1\right) x^2+\left((t+2)^{1 / 0}-1\right) x+\left((t+2)^{2...

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Q JEE MAIN 2023

$\operatorname{Lim}_{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \cdots \cdots\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}$ is equal to

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Q JEE MAIN 2025

If $\lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p$, then $96 \log _e p$ is equal to $\_\_\_\_$

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Q JEE MAIN 2025

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to

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Q JEE MAIN 2025

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f^{\prime}(2)=4$. Let $\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=\mathbf{e}^\alpha$. Then...

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Q JEE MAIN 2025

If $\lim _{x \rightarrow 0} \frac{\cos (2 x)+a \cos (4 x)-b}{x^4}$ is finite, then $(a+b)$ is equal to :

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Q JEE MAIN 2023

$\lim _{x \rightarrow 0}\left(\left(\frac{1-\cos ^2(3 x)}{\cos ^3(4 x)}\right)\left(\frac{\sin ^3(4 x)}{\left(\log _e(2 x+1)\right)^5}\right)\right)$ is equal to $\_\_\_\_$

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Q JEE MAIN 2024

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots 10}{x^2}\right)$ is

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Q JEE MAIN

Let a circle passing through (2,0) have its centre at the point (h,k). Let $\left(x_c, y_c\right)$ be the point of...

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Q JEE-Main 2024

Let $\mathrm{f}:(-\infty, \infty)-\{0\} \rightarrow \mathrm{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $...

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Q JEE MAIN 2024

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}$,...

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Q JEE MAIN 2024

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function given by $$ f(x)=\left\{\begin{array}{cl} \frac{1-\cos 2 x}{x^2} & , x0 \end{array}\right. $$...

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Q JEE-Main 2024

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $L$ and R...

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Q JEE MAIN 2024

$\lim _{x \rightarrow 0} \frac{\mathrm{e}^{2|\sin x|}-2|\sin x|-1}{x^2}$

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Q JEE MAIN 2024

If $a=\lim _{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b=\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $...

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Q JEE MAIN 2025

If $\lim _{\mathrm{t} \rightarrow 0}\left(\int_{0}^{1}(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{\mathrm{t}}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to $\_\_\_\_$ .

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Q JEE MAIN 2025

Let $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^{3}\left(x / 2^{r+1}\right)}{1-\tan ^{2}\left(x / 2^{r+1}\right)}\right)$. Then $...

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Q JEE MAIN 2025

$...

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Q JEE MAIN 2025

If $...

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Q JEE MAIN 2025

For $\alpha ,\beta ,\gamma \in R$ , if $...

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Q JEE MAIN 2025

$\lim _{x \rightarrow \infty} \frac{\left(2 x^{2}-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^{2}+5 x+4\right) \sqrt{(3 x+2)^{x}}}$ is equal to :

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Q JEE MAIN 2025

Let [t] be the greatest integer less than or equal to t. Then the least value of $p\in \mathbb{N}$ for...

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Q JEE MAIN 2025

$...

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Q JEE MAIN 2025

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left( {\frac{1}{x}} \right) = \frac{{35}}{{3x}} - \frac{5}{2}$ ....

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Q JEE MAIN 2025

If $\lim _{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :

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