Chapter 1.1, Problem 58E

(a)

To determine

To draw: The graph of $f$ .

Answer to Problem 58E

The graph is shown:

  

Explanation of Solution

Given information: The functions are:

  $\begin{array}{c}c=2,\\ f(x)=\left\{\begin{array}{c}3-x,x<2\\ 2,x=2\\ x/2,x>2\end{array}\right.\end{array}$

Calculation:

The provided function is graphed. Be aware that some graphing tools might not display the hole and discontinuity.

  

(b)

To determine

To find: The value of $\underset{x\to c^{+}}{\lim }f(x)$ and

  $\underset{x\to c^{-}}{\lim }f(x)$ .

Answer to Problem 58E

  $\underset{x\to 2^{+}}{\lim }f(x)=1$ and

  $\underset{x\to 2^{-}}{\lim }f(x)=1$

Explanation of Solution

Given information: The functions are:

  $\begin{array}{c}c=2,\\ f(x)=\left\{\begin{array}{c}3-x,x<2\\ 2,x=2\\ x/2,x>2\end{array}\right.\end{array}$

Calculation:

Put $x=2$ and calculate the limit:

  $\begin{array}{c}\underset{x\to 2^{+}}{\lim }\frac{x}{2}=\frac{2}{2}\\ =1\end{array}$

Put, $x=2\text{ into }3-x$ and get:

  $\begin{array}{c}\underset{x\to 2^{-}}{\lim }3-x=3-2\\ =1\end{array}$

Therefore the required value of $\underset{x\to c^{+}}{\lim }f(x)$ and $\underset{x\to c^{-}}{\lim }f(x)$ are:

  $\underset{x\to 2^{+}}{\lim }f(x)=1$ and

  $\underset{x\to 2^{-}}{\lim }f(x)=1$

(c)

To determine

To find: The limit $\underset{x\to c}{\lim }f(x)$ exists or no.

Answer to Problem 58E

  $\underset{x\to c}{\lim }f(x)$ exists and $=1$ .

Explanation of Solution

Given information: The functions are:

  $\begin{array}{c}c=2,\\ f(x)=\left\{\begin{array}{c}3-x,x<2\\ 2,x=2\\ x/2,x>2\end{array}\right.\end{array}$

Calculation:

Because it is identical to the limit as xx approaches 2 from both the right and the left, which is 1, the value of the limit as $x$ approaches 2.

Therefore the required $\underset{x\to c}{\lim }f(x)$ exists and $=1$ .