Indeterminate Forms

If $\underset{x\to 0}{\lim }\frac{1-cosx.cos2x.cos3x\dots \dots \dots \cos nx}{x^2}$ has the value equal to$253$, find the value of $n$ is equal to (where $n\in N$)

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

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

Evaluate: $\underset{x\to 0}{\lim }{\left(\frac{1}{x}\right)}^{\sin x}$.

Evaluate: $\underset{x\to \infty }{\lim }{\left(x\right)}^{e^{-x}}$.

Evaluate: $\underset{x\to \infty }{\lim }{\left(1+x^2\right)}^{e^{-x}}$.

Evaluate: $\underset{x\to \infty }{\lim }{x}^{\left(1/x\right)}$.

Evaluate: $\underset{x\to 2}{\lim }{\left(x-2\right)}^{\left(x-2\right)}$.

Evaluate: $\underset{x\to 0}{\lim }\left[\tan x\left(\ln x\right)\right]$.

Evaluate: $\underset{x\to 0}{\lim }{x}^{\tan x}$

Evaluate: $\underset{x\to 0}{\lim }{x}^{\sin x}$

Evaluate: $\underset{x\to 0}{\lim }{x}^x$.

Evaluate: $\underset{x\to -\infty }{\lim }x{e}^x$.

Evaluate: $\underset{x\to 0}{\lim }x\cot x$.

Evaluate: $\underset{x\to 0^{+}}{\lim }x\ln x$.

$\underset{x\to \infty }{\lim } \frac{2x-{\ln }^{2}x}{\sqrt{x} \ln x+x}$ is equal to

The value of $\underset{x\to 0}{\lim }\frac{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos x\right)}^{\frac{1}{3}}}{{\left(\cos x\right)}^{\frac{1}{2}}-{\left(\cos x\right)}^{\frac{1}{5}}}$ is equal to

$\underset{x\to \infty }{\lim } \frac{2x-{\ln }^{2}x}{\sqrt{x} \ln x+x}$ is equal to

Let $f\left(x\right)={\left(\frac{x-1+\cos x}{x}\right)}^{\frac{1}{x}}$ be a continuous function at $x=0$. If $f\left(0\right)=e^{\lambda }$, then the value of $\left|\right[\lambda \left]\right|$, where $[ ]$ denotes greatest integer function, is

If $f\left(2\right)=4,f^{\text{'}}\left(2\right)=1$ then find $\underset{x\to 2}{\lim }\left[\frac{xf\left(2\right)-2f\left(x\right)}{x-2}\right]$