Chapter 3.3, Problem 49E

(a)

To determine

To find: the right end behavior model

Answer to Problem 49E

The function value is $\frac{\pi }{2}$ for right end behavior.

Explanation of Solution

Given information :

The function is $y={\tan }^{-1}x$ .

The domain of function $y={\tan }^{-1}x$ is $\left(-\infty ,\infty \right)$ .

The range of function $y={\tan }^{-1}x$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

For right end behavior, $x\to \infty$ .

  $\begin{array}{c}\underset{x\to \infty }{\lim }y=\underset{x\to \infty }{\lim }{\tan }^{-1}x\\ =\frac{\pi }{2}\end{array}$

As $x$ approach to $\infty$ , the function tends to $\frac{\pi }{2}$ .

(b)

To determine

To find: the left end behavior model

Answer to Problem 49E

The function value is $-\frac{\pi }{2}$ for left end behavior.

Explanation of Solution

Given information :

The function is $y={\tan }^{-1}x$ .

The domain of function $y={\tan }^{-1}x$ is $\left(-\infty ,\infty \right)$ .

The range of function $y={\tan }^{-1}x$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

For left end behavior, $x\to -\infty$ .

  $\begin{array}{c}\underset{x\to -\infty }{\lim }y=\underset{x\to -\infty }{\lim }{\tan }^{-1}x\\ =-\frac{\pi }{2}\end{array}$

As $x$ approach to $-\infty$ , the function tends to $-\frac{\pi }{2}$ .

(c)

To determine

To find: horizontal tangents for the function if exist.

Answer to Problem 49E

No horizontal tangents

Explanation of Solution

Given information :

The function is $y={\tan }^{-1}x$ .

The domain of function $y={\tan }^{-1}x$ is $\left(-\infty ,\infty \right)$ .

The range of function $y={\tan }^{-1}x$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

If function have horizontal tangents, then slope of curve is $\frac{dy}{dx}=0$ .

Differentiate $y={\tan }^{-1}x$ with respect to $x$ ,

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}{\tan }^{-1}x\\ =\frac{1}{1+x^2}\end{array}$

The slope of curve is not equal to zero.

Hence, no horizontal tangent line exists.