Chapter 11.1, Problem 80E

a.

To determine

To find: To find the function $f\left(0\right)$ exists in the given graph.

Answer to Problem 80E

The function $f\left(0\right)$ exists in the given graph.

Explanation of Solution

Given:

  

From the graph we can observe that there is a hole and a solid dot at $x=0$ .

The solid dot means $f\left(0\right)$ is been defined.

  $f\left(0\right)=1$

Therefore we can conclude that the function $f\left(0\right)$ is defined and exists.

b.

To determine

To find: To find the function $\underset{x\to 0}{\lim }f\left(x\right)$ exists in the given graph.

Answer to Problem 80E

The function $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist in the given graph.

Explanation of Solution

Given:

  

From the observation of the graph,

From the left hand side, $\underset{x\to 0}{\lim }f\left(x\right)$ approaches $f\left(x\right)\to 4$ at $x\to 0$ .

From the right hand side, $\underset{x\to 0}{\lim }f\left(x\right)$ approaches $f\left(x\right)\to 1$ at $x\to 0$ .

The function $f\left(x\right)$ approaches two different values at $x\to 0$ .

Therefore the function $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

c.

To determine

To find: To find the function $f\left(2\right)$ exists in the given graph.

Answer to Problem 80E

The function $f\left(2\right)$ is undefined in the given graph.

Explanation of Solution

Given:

  

From the graph we can observe that there is a hole at $x=2$ .

The hole at $x=2$ means $f\left(2\right)$ is been undefined.

Therefore we can conclude that the function $f\left(2\right)$ is undefined.

d.

To determine

To find: To find the function $\underset{x\to 2}{\lim }f\left(x\right)$ exists in the given graph.

Answer to Problem 80E

The function $\underset{x\to 2}{\lim }f\left(x\right)$ exists in the given graph.

Explanation of Solution

Given:

  

From the graph we can observe that,

From the left hand side and right hand side $x\to 2$ .

Therefore we can conclude that the function $\underset{x\to 2}{\lim }f\left(x\right)$ exists.