Q. 1 TB - Calculus

Question

Determine whether each function approaches 0, approaches a nonzero real number, or becomes infinite as x approaches each indicated value.

$$\begin{gathered} f\left(x\right)=csc x, \mathrm{with} x\to 0 \mathrm{and} x\to \pi . \\ f\left(x\right)={\tan }^2 x, \mathrm{with} x\to 0 \mathrm{and} x\to \pi . \\ f\left(x\right)={\sin }^{-1} x, \mathrm{with} x\to 0 \mathrm{and} x\to 1. \\ f\left(x\right)={\tan }^{-1}\sqrt{x}, \mathrm{with} x\to 0 \mathrm{and} x\to 3. \end{gathered}$$

Step-by-Step Solution

Answer

Function $f\left(x\right)=csc x$approaches $\infty$as $x\to 0$and $x\to \pi .$

Function $f\left(x\right)={\tan }^2 x$approaches $0$as $x\to 0 \mathrm{and} x\to \pi .$

Function $f\left(x\right)={\sin }^{-1} x$approaches $0$as $x\to 0$and approaches $\frac{\mathrm{\pi }}{2}$as$x\to 1.$

Function $f\left(x\right)={\tan }^{-1}\sqrt{x}$approaches $0$ as $x\to 0$and approaches $\frac{\mathrm{\pi }}{3}$as $x\to 3.$

Step 1. Given information.

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

Step 2. Limits of the first function.

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

Check the function as $x\to 0.$

$\underset{x\to 0}{\lim }csc x=\infty$

Check the function as $x\to \mathrm{\pi }.$

$\underset{x\to \mathrm{\pi }}{\lim }csc x=\infty$

Step 3. Limits of the second function.

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

Check the function as $x\to 0.$

$\underset{x\to 0}{\lim }{\tan }^2 x=0$

Check the function as $x\to \pi .$

$\underset{x\to \mathrm{\pi }}{\lim }{\tan }^2 x=0$

Step 4. Limits of the third function.

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

Check the function as $x\to 0.$

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

Check the function as $x\to 1.$

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

Step 5. Limits of the fourth function.

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

Check the function as $x\to 0.$

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

Check the function as $x\to 3.$

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