Chapter 11.3, Problem 79E

Solution

a)

To draw: The graph of the function.

The graph for the function is:

  

Given information:

The given function is $f\left(x\right)=\left\{\begin{array}{c}\cos x\text{if}-\pi \le x<0\\ -\cos x\text{if}0\le x\le \pi \end{array}\right.$

Calculation:

Plot the graph for $y=\cos x$ . Place a solid dot at $x=-\pi$ and an open dot at $x=0$ .

Plot the graph for $y=-\cos x$ , a reflection across the $x$ -axis of the graph $y=\cos x$ .

Plot the graph for $y=\cos x$ between $-\pi \le x<0$ and $y=-\cos x$ between $0\le x\le \pi$ .

  

b)

To find: The point $c$ in the domain of $f$ where $\underset{x\to c}{\lim }f\left(x\right)$ exist.

The point $c$ in the domain of $f$ where $\underset{x\to c}{\lim }f\left(x\right)$ exist is $\left(-\pi ,0\right)\cup \left(0,\pi \right)$ .

Given information:

The given function is $f\left(x\right)=\left\{\begin{array}{c}\cos x\text{if}-\pi \le x<0\\ -\cos x\text{if}0\le x\le \pi \end{array}\right.$

Calculation:

The $\underset{x\to c}{\lim }f\left(x\right)$ exists and is equal to $L$ if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=L$ .

This occurs for all $x$ values $\left(-\pi ,0\right)\cup \left(0,\pi \right)$ .

c)

To find: The point where only the left hand limit exists.

The point where only the left hand limit exists is $x=\pi$ .

Given information:

The given function is $f\left(x\right)=\left\{\begin{array}{c}\cos x\text{if}-\pi \le x<0\\ -\cos x\text{if}0\le x\le \pi \end{array}\right.$

Calculation:

Only the left-hand limit at point $c$ exists when there is no graph to the right of $x=c$ or if the right-hand limit is $\infty \text{or}-\infty$ .

Therefore, point where only the left hand limit exists is $x=\pi$ .

d)

To find: The point where only the right hand limit exists.

The point where only the right hand limit exists is $x=-\pi$ .

Given information:

The given function is $f\left(x\right)=\left\{\begin{array}{c}\cos x\text{if}-\pi \le x<0\\ -\cos x\text{if}0\le x\le \pi \end{array}\right.$

Calculation:

Only the right-hand limit at point $c$ exists when there is no graph to the left of $x=c$ or if the left-hand limit is $\infty \text{or}-\infty$ .

Therefore, point where only the right hand limit exists is at $x=-\pi$ .