Basics of Limits

Find the limit.

$\underset{x\to 2}{\lim }\left(\frac{\sqrt{1-\cos \left\{2\left(x-2\right)\right\}}}{x-2}\right)$

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 $\text{f}\left(\text{n, }\theta \right)=\underset{\text{r}=1}{\overset{\text{n}}{\Pi }}\left(1-{\text{tan}}^2\frac{\theta }{2^{\text{r}}}\right)$, then compute $\underset{\text{n}\to \infty }{\text{Lim}}\text{ f }\left(\text{n}, \theta \right)$       { given $\theta \ne 0$ }

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

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

$\underset{x\to 0}{\lim }x\sec x$ is

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

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

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

The limit $\underset{x\to 1}{\lim }\frac{\sqrt{1-\cos 2(x-1)}}{x-1}\_\_\_\_\_\_\_\_\_$

$\underset{x\to 1}{\lim }\left(\left(1-x\right)+\left[x-1\right]+\left|1-x\right|\right)$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$

Let $f\left(x\right)=x\left[\frac{1}{x}\right]$ for $x(\ne 0)\in R,$ where for each $t\in R, \left[t\right]$ denotes the greatest integer less than or equal to $t.$ Then,

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

Evaluate:$\underset{x\to 0}{\lim }\frac{e^{\sin x}-1}{x}=$

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

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

$\underset{x\to 0}{\lim }\frac{\sin 5x}{\tan 3x}=$

$\underset{x\to 1}{\lim }\frac{\mathrm{sin\pi }x}{x-1}=$