91Ó°ÊÓ

Given functions with limits are the following.
$$\begin{gathered} f\left(x\right)=cscx,\mathrm{with}x→0\mathrm{and}x→\mathrm{π}. \\ f\left(x\right)={\tan }^{2}x,\mathrm{with}x→0\mathrm{and}x→\mathrm{π}. \\ f\left(x\right)={\sin }^{-1}x,\mathrm{with}x→0\mathrm{and}x→1. \\ f\left(x\right)={\tan }^{-1}\sqrt{x},\mathrm{with}x→0\mathrm{and}x→3. \end{gathered}$$

Take function $f\left(x\right)=cscx.$

Check the function as $x→0.$

$\underset{x→0}{\lim }cscx=∞$

Check the function as$x→\mathrm{π}.$

$\underset{x→\mathrm{π}}{\lim }cscx=∞$

Take function$f\left(x\right)={\tan }^{2}x.$

Check the function as$x→0.$

role="math" localid="1648721836526" $\underset{x→0}{\lim }{\tan }^{2}x=0$

Check the function as $x→\mathrm{π}.$

$\underset{x→\mathrm{π}}{\lim }{\tan }^{2}x=0$

Take function$f\left(x\right)={\sin }^{-1}x.$

Check the function as$x→0.$

$$\begin{gathered} \underset{x→0}{\lim }f\left(x\right)=\underset{x→0}{\lim }{\sin }^{-1}x \\ \underset{x→0}{\lim }f\left(x\right)=0 \end{gathered}$$

Check the function as $x→1.$

$$\begin{gathered} \underset{x→1}{\lim }f\left(x\right)=\underset{x→0}{\lim }{\sin }^{-1}x \\ \underset{x→1}{\lim }f\left(x\right)=\frac{\mathrm{π}}{2} \end{gathered}$$

Take function$f\left(x\right)={\tan }^{-1}\sqrt{x}.$

Check the function as$x→0.$

$$\begin{gathered} \underset{x→0}{\lim }f\left(x\right)=\underset{x→0}{\lim }{\tan }^{-1}\sqrt{x} \\ \underset{x→0}{\lim }f\left(x\right)=0 \end{gathered}$$

Check the function as $x→3.$

$$\begin{gathered} \underset{x→3}{\lim }f\left(x\right)=\underset{x→0}{\lim }{\tan }^{-1}\sqrt{x} \\ \underset{x→3}{\lim }f\left(x\right)=\frac{\mathrm{π}}{3}. \end{gathered}$$