Chapter 1.1, Problem 63E

(a)

To determine

To find: The graph of $f$ .

Answer to Problem 63E

The graph is shown:

  

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\sqrt{1-x^2},\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}\right.$

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 63E

All $c\in (0,1)\cup (1,2)$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\sqrt{1-x^2},\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}\right.$

Calculation:

The domain $f$ is $(-2\pi ,2\pi )$ .The limit will not exists.

At the end points

  $\begin{array}{l}x=1,\\ x=2\end{array}$

The limit only exists on only one side at this point.

At $x=1$ as the left and right hand limits are different at the point.

Therefore the required limits exits in the interval $(0,1)\cup (1,2)$ .

(c)

To determine

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

Answer to Problem 63E

  $c=2$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\sqrt{1-x^2},\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}\right.$

Calculation:

At the point $x=2$ only left hand limit is exists as the function is not defined for points to the right of the point.

Therefore the required point that c does only the left hand limits exits is $c=2$ .

(d)

To determine

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

Answer to Problem 63E

  $c=0$

Explanation of Solution

Given information: The function is:

  $f(x)=\left\{\begin{array}{c}\sqrt{1-x^2},\le x<1\\ 1,1\le x<2\\ 2,x=2\end{array}\right.$

Calculation:

At $x=0$ only right hand limit is exists as the function is not defined for points to the left of the point.

Therefore the required point that c does only the right hand limits exits is $c=0$ .