Indeterminate Forms

Let $a_1>a_2>a_3>\dots >a_n>1; p_1>p_2>p_3>\dots >p_n>0;$ such that $p_1+p_2+p_3+\dots +p_n=1$. Also $F\left(x\right)={\left(p_1{a}_1^x+p_2{a}_2^x+\dots +p_n{a}_n^x\right)}^{1/x},$ then $\underset{x\to \infty }{\lim }F\left(x\right)$ equals

Let $f\mathit{: }R\mathit{\to }R$ be such that $f\left(1\right)=3$ and $f^{\text{'}}\left(1\right)=6$. Then $\underset{x\to 0}{\mathrm{Lim}}{\left(\frac{f\left(1+x\right)}{f\left(1\right)}\right)}^{1/x}$

$\underset{x\to 0}{\lim }{x}^x$ is equal to

Evaluate:

$\underset{x\to \infty }{\lim }{\left(1+x\right)}^{\frac{1}{x}}$

Evaluate:

$\underset{x\to \alpha }{\lim }{\left(x-\alpha \right)}^{\left(x-\alpha \right)}$

The value of $\underset{x\to \infty }{\lim }{\left(\frac{\mathrm{\pi }}{2}-{\tan }^{-1}x\right)}^{\frac{1}{x}}$ is

$\underset{x\to 2}{\lim }\left[\frac{1}{x-2}-\frac{2(2x-3)}{x^3-3x^2+2x}\right]$ is equal to

$\underset{x\to \infty }{\lim }[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]$ is equal to

The value of $\underset{x\to 0}{\lim }{\log }_e{\left(\sin x\right)}^{\tan x}$ is

Evaluate:

$\underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }{\left(\tan x\right)}^{\tan 2x}$

Evaluate:

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

Evaluate:

$\underset{x\to 1^{-}}{\lim }{\left(1-x^2\right)}^{\frac{{\displaystyle 1}}{\log \left(1-x\right)}}$

Evaluate:

$\underset{x\to 1^{-}}{\lim }\log \left(1-x\right)\cot \frac{\mathrm{\pi }}{2}x$

The limit $\underset{x\to 0}{\lim }{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}}$ is of the form $e^{\frac{m}{n}}$, then $m+n=$

If $\underset{x\to a}{\lim }{\left(2-\frac{a}{x}\right)}^{{\tan }^{\frac{\mathrm{\pi }x}{2a}}}$ is in the form of $e^{\frac{m}{\mathrm{\pi }}}$, then $m=$

Evaluate:

$\underset{x\to \infty }{\lim }{2}^x\sin \frac{a}{2^x}, a\ne 0$.

Evaluate:

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

Evaluate:

$\underset{x\to \frac{{\mathrm{\pi }}^{-}}{2}}{\lim }{\left(\sec x\right)}^{\cot x}$

Evaluate:

$\underset{x\to 0^{+}}{\lim }{\left(\cot x\right)}^{\sin 2x}$

Evaluate:

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