Limits and Cont...

$$ \displaystyle \lim _{n \rightarrow \infty} \sum_{x=1}^{20} \cos ^{2 n}(x-10) $$ is equal to

$$\displaystyle  \lim _{x \rightarrow 0} \dfrac{\sin \left(x^{2}\right)}{\ln \left(\cos \left(2 x^{2}-x\right)\right)} $$ is equal to

$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^{2}} $$ is equal to

$$\displaystyle \lim _{x \rightarrow 0} \dfrac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]$$ is equal to

If $$y^{r}=\dfrac{n !^{n+r-1} C_{r-1}}{r^{n}},$$ where $$n=k r(k \text { is constant }),$$ then $$\operatorname{lim}_{r\rightarrow\infty} y$$ is equal to

$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\cos (\tan x)-\cos x}{x^{4}} $$ is equal to

The value of $$ \displaystyle \lim _{x \rightarrow a} \sqrt{a^{2}-x^{2}} \cot \dfrac{\pi}{2} \sqrt{\dfrac{a-x}{a+x}} $$ is

The value of $$ \displaystyle \lim _{n \rightarrow \infty}\left[\dfrac{1}{n}+\dfrac{e^{1 / n}}{n}+\dfrac{e^{2 / n}}{n}+\ldots .+\dfrac{e^{(n-1) / n}}{n}\right] $$ is

If function $$f(x)=\dfrac{x^2-9}{x-3}$$ is continuous at $$x=3$$, then value of $$(3)$$ will be:

If $$f(x)=\begin{cases} \begin{matrix} \dfrac{\log (1+mx)- \log (1-nx)}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} k; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=0$$ then the value of $$k$$ will be: