Basics of Limits

Let $f\left(x\right)=\left\{\begin{array}{l}\sin x; x\in Z\\ 0; \mathrm{otherwise}\end{array}\right.$ and $$\begin{gathered} g\left(x\right)=\left\{\begin{array}{l}{x}^2+1; x\ne 0,2 \\ 4; x=0\\ 5; x=2\end{array}\right. \end{gathered}$$, then

Solve the limit $\underset{x\to 0}{\lim }\left[\cos x\right]$ {where $\left[.\right]$ denotes the greatest integer function} 

Prove that $\underset{x\to 3}{\lim }\frac{x-3}{\sqrt{\left|x^2-9\right|}}=0$

Prove that $\underset{x\to 2^{-}}{\lim }\frac{\left|x-2\right|}{x-2}=-1$

$\underset{n\to \infty }{\lim }\left[\frac{\left(1+2+3+...+n\right)\left(1^2+2^2+3^2+...+n^2\right)}{n\left(1^3+2^3+3^3+...+n^3\right)}\right]=$

$\underset{x\to 0}{\lim }\left(\frac{\sin x}{\left|x\right|}\right)=$

$\underset{n\to \infty }{\lim }\left(\frac{1^2+1}{n^3}+\frac{2^2+2}{n^3}+\frac{3^2+3}{n^3}+....+\frac{n^2+n}{n^3}\right)=$

$\underset{x\to \infty }{\lim }\left(\frac{5x^3+2x-1}{4x^5-2x^2+x-3}\right)=$

If $f\left(x\right)=\frac{x^2-3x+2}{x^2-7x+12}$, then which of the following exists?

$\left(\mathrm{i}\right) \underset{x\to -\infty }{\lim }{\sin }^{-1}\left(f\left(x\right)\right)$

$\left(\mathrm{ii}\right) \underset{x\to \infty }{\lim }{\cos }^{-1}\left(f\left(x\right)\right)$

If $\left[·\right]$ denotes the greatest integer function, then $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{5\sin \left[\cos x\right]}{\left[\cos x\right]+2}\right)$ is:

$\underset{\mathrm{x}\to 0^{+}}{\lim }\left(\left(x^{x^x}\right)-x^x\right)$ is

$\underset{x\to 0}{\lim }\left(1+x{e}^{-\frac{1}{x^2}}\sin \left(\frac{1}{x^4}\right)\right)=$

$\underset{x\to 2}{\lim }\frac{3x-9}{x-2}=$

$\underset{x\to 0}{\lim }{x}^{\frac{3}{2}}\left(x>0\right)=$

If $A_i=\frac{x-a_i}{\left|x-a_i\right|},i=1,2,3,\dots .,n$ and $a_1<a_2<a_3<\dots ..<a_n$, $\underset{x\to a_m}{\lim }\left(A_1{A}_2\dots .A_n\right),1\le m\le n$

$\underset{x\to 1}{\lim }\left(\left(1-x\right)+\left[x-1\right]+\left|1-x\right|\right)$ 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,

$\underset{x\to 4}{\lim }\frac{x-4}{\left|x-4\right|}=$

Let $f\left(x\right)=(1+\sin x)e^x$. Then the value of $\underset{x\to 0}{\lim }\left[f\left(x\right)\right]$ is, where $[.]$ denotes greatest integer function

For next two question please follow the same

If $\underset{\text{x}\to 0^{+}}{\text{lim}}\text{f}\left(\text{x}\right)=$ finite, where $\text{f}\left(\text{x}\right)=\frac{\text{sin x}+{\text{ae}}^{\text{x}}+{\text{be}}^{-\text{x}}+\text{c ln}\left(1+\text{x}\right)}{{\text{x}}^3}$ and $\mathrm{a},\mathrm{b},\mathrm{c}$ are real numbers.

Then the value of $\mathrm{a}$ is