If $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\mathrm{ }\frac{{\mathrm{e}}^{\left[\mathrm{x}\right]+\left|\mathrm{x}\right|}-2}{\left[\mathrm{x}\right]+\left|\mathrm{x}\right|}$ then

Important Questions on Limits

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let $\underset{x\to 0}{\lim }\frac{x{5}^x-x}{1-\cos x}$ is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{x\to -\infty }{\lim }\frac{3\left|x\right|-x}{\left|x\right|-2x}-\underset{x\to 0}{\lim }\frac{\log \left(1+x^3\right)}{{\sin }^{3}x}=$

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let $0<\alpha <\beta <1$. Then, $\underset{n\to \infty }{\lim }{\int }_{1/(k+\beta )}^{1/(k+\alpha )}\frac{\mathrm{d}x}{1+x}$ is

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of $\underset{x\to 0}{\lim }\frac{\left(1-\cos 2x\right)\left(3+\cos x\right)}{x\tan 4x}$ is equal to 

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{x\to 0}{\lim }\frac{\sin \left({\mathrm{\pi cos}}^{2}x\right)}{x^2}$ is equal to 

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

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

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{n\to \infty }{\lim }\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+\dots +(n \text{terms})\right)=$

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

If $\underset{x\to 0}{\lim }\frac{x^a{\sin }^{b}x}{\sin x^c},a,b,c,\in R~\left\{0\right\}$ exists and has non-zero value, then

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

If $\underset{x\to \infty }{\lim }{\left(1+\frac{a}{x}-\frac{4}{x^2}\right)}^{2x}=e^3$, then $a$ is equal to 

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

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

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let $f : \mathrm{R}\to \mathrm{R}$ satisfy the equation $f(x+y)=f\left(x\right)·f\left(y\right)$ for all $x,y\in R$ and $f\left(x\right)\ne 0$ for any $x\in R$. If the function $f$ is differentiable at $x=0$ and $f^{\text{'}}\left(0\right)=3$, then $\underset{h\to 0}{\lim }\frac{1}{h}\left(f\left(h\right)-1\right)$ is equal to ___ .

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{x\to 0}{\lim }\frac{{\mathrm{e}}^{x^2}-\cos x}{{\sin }^2 x}$is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let $f\left(x\right)=\frac{1}{3}x\sin x-\left(1-\cos x\right)$. The smallest positive integer $k$ such that $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^k}\ne 0$ is

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

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

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

Let for all $x>0, f\left(x\right)=\underset{n\to \infty }{\lim }n\left(x^{1/n}-1\right),$ then

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

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

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{\theta \to \frac{\pi }{4}}{\lim }\frac{\sqrt{2}-\cos \theta -\sin \theta }{(4\theta -\pi {)}^2}$ is equal to

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

If $f\left(x\right)=\left|\begin{array}{lll}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& x\end{array}\right|,$ $x\mathit{\in }\left(\mathit{-}\frac{\mathit{\pi }}{\mathit{2}}\mathit{,}\frac{\mathit{\pi }}{\mathit{2}}\right)\mathit{,}$ then $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}$ is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{x\to 0}{\lim }\frac{1-\cos mx}{1-\cos nx}=$

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\underset{x\to 0}{\lim }\frac{\left(1-cos2x\right)\left(3+cosx\right)}{xtan4x}=$