Introduction to Limits

If $\underset{x\to \infty }{\lim }\frac{\left(e^{\mu x}+5\right)}{\left(e^{100x}+7\right)}$ exists, then sum of all possible positive integral values of $\mu$ is

It is given that $\underset{x\to 0}{\lim }\frac{\frac{x^2}{{\left(a+x^r\right)}^{{\displaystyle \frac{1}{p}}}}}{1-\cos \mathit{x}}=\ell$. If $P=3$ and $\ell =1$, then the value of $a$ is equal to

If $\underset{x\to 0}{\lim }\frac{\sin \left(nx\right)[(a-n)nx-\tan x]}{x^2}=0, (n>0)$ then the value of $\text{'}a\text{'}$ is equal to

Show that $\underset{x\to 0}{\lim } \frac{{10}^x-2^x-5^x+1}{x^2}=\log 2.\log 5$

Evaluate:

$\underset{x\to \infty }{\lim }\frac{x^4+3x^3-2x^2+5}{x^3-3x^2+2x-1}$

If $f\left(x\right)=\left\{\begin{array}{cc}x+2,& x<1\\ 4x-1,& 1\le x\le 3\\ x^2+5,& x>3\end{array}\right.$then find the value of $\underset{x\to 1^{+}}{\lim }f\left(x\right)$, $\underset{x\to 1^{-}}{\lim }f\left(x\right)$, $\underset{x\to 3^{+}}{\lim }f\left(x\right)$ and $\underset{x\to 3}{\lim }f\left(x\right)$

Evaluate the $\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{1}{n^3}\sum _{k=1}^n\left[k^{2}x\right]$, (where $\left[x\right]$ denotes greatest integer function less than or equal to $x$).

Evaluate:

$\underset{x\to 0}{\lim }\frac{e^x+\tan x-1}{\log \left(1+x\right)}$

Evaluate:

$\underset{x\to 0}{\lim }\frac{e^{3x}-1}{\log \left(1+5x\right)}$

Evaluate:

$\underset{x\to 0}{\lim }\frac{\log \left(1+\sin x\right)}{x}$

Evaluate:

$\underset{x\to 0}{\lim }\frac{\log \left(1+4x\right)}{x}$

Evaluate:

$\underset{x\to 0}{\lim }\frac{e^{\sin px}-e^{\sin qx}}{x}$

$\underset{x\to 0}{\lim }\frac{a^{x^2}-b^{x^2}}{x^2}=$

$\underset{x\to 0}{\lim }\frac{e^{2x}-e^{-2x}}{x}=$

$\underset{x\to 0}{\lim }\frac{e^{13x}-e^{7x}}{x}=$

Evaluate:

$\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{\sqrt{1+x\sin x}-\cos x}$

Evaluate:

$\underset{x\to \frac{1}{2}}{\lim }\frac{\sin \left({\displaystyle \frac{\mathrm{\pi x}}{2}}\right)-\cos \left({\displaystyle \frac{\mathrm{\pi x}}{2}}\right)}{{\displaystyle \frac{1}{2}}-x}$

Evaluate:

$\underset{x\to y}{\lim }\frac{{\sin }^{2}x-{\sin }^{2}y}{x^2-y^2}$

Evaluate:

$\underset{x\to \frac{\mathrm{\pi }}{3}}{\lim }\frac{\sin \left(x-{\displaystyle \frac{\mathrm{\pi }}{3}}\right)}{1-2\cos x}$