Chapter 1.1, Problem 62E

(a)

To determine

To find: The graph of $f$ .

Answer to Problem 62E

The graph is shown:

  

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\cos x,-\pi \le x<0\\ \sec x,0\le x\le \pi \end{array}\right.$

Graph:

The graph of the given function is shown below:

  

(b)

To determine

To find: The points of c in the domain of $f$ does $\underset{x\to c}{\lim }f(x)$ exists.

Answer to Problem 62E

  $c\in \left(-\pi ,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right)$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\cos x,-\pi \le x<0\\ \sec x,0\le x\le \pi \end{array}\right.$

Calculation:

From part (a):

  $x=\frac{\pi }{2}$

At $x=-\pi$ only the right-hand limit exist since it is an endpoint. Therefore the $\underset{x\to -\pi }{\lim }f(x)$ will not exist since it doesn't have a left-hand limit.

Therefore the limit will exist for c in the interval $\left(-\pi ,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right)$ .

(c)

To determine

To find: The points that c does only the left hand limits exits.

Answer to Problem 62E

  $c=\pi$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\cos x,-\pi \le x<0\\ \sec x,0\le x\le \pi \end{array}\right.$

Calculation:

At $x=\pi$ only left hand limit is exists as the function is not defined for points to the right of the point.

(d)

To determine

To find: The points that c does only the right hand limits exits.

Answer to Problem 62E

  $c=-\pi$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\cos x,-\pi \le x<0\\ \sec x,0\le x\le \pi \end{array}\right.$

Calculation:

At $x=-\pi$ only right hand limit is exists as the function is not defined for points to the left of the point.