Chapter 11.1, Problem 80E

(a)

To determine

To find: whether the value of the $f\left(0\right)$ quantity exists if it does solve it if not explain.

Answer to Problem 80E

  $f\left(0\right)=1$ .

Explanation of Solution

Given:

  

Concept used:

If there is highlight dot in the graph means the limit exist on the point and if there is a hole in the graph at the value that x is approaching, with no other point for a different value of the function, then the limit does still exist.

Calculation:

Simply look at where the graph has filled in bubble:

From the given graph it is clear that:

  $f\left(0\right)=1$ .

Hence, yes $f\left(0\right)=1$ .

(b)

To determine

To find: whether the value of the $\underset{x\to 0}{\lim }f\left(x\right)$ quantity exists if it does solve it if not explain.

Answer to Problem 80E

The $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t not exist.

Explanation of Solution

Given:

  

Concept used:

If the function is not continuous then limit on the point doesn’t not exist.

Calculation:

the function is not continuous, and there is a break in the graph. The limits did not exist.

Hence, $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t not exist.

(c)

To determine

To find: whether the value of the $f\left(2\right)$ quantity exists if it does solve it if not explain.

Answer to Problem 80E

  $f\left(2\right)$ is hole no value.

Explanation of Solution

Given:

  

Concept used:

If there is highlight dot in the graph means the limit exist on the point and if there is a hole in the graph at the value that x is approaching, with no other point for a different value of the function, then the limit does still exist.

Calculation:

Holes in the graphs do not have values.

Since, $f\left(2\right)$ lies in the holes therefore it does not have exact values.

Hence, $f\left(2\right)$ is hole no value.

(d)

To determine

To find: whether the value of the $\underset{x\to 2}{\lim }f\left(x\right)$ quantity exists if it does solve it if not explain.

Answer to Problem 80E

  $\underset{x\to 2}{\lim }f\left(x\right)=3$ .

Explanation of Solution

Given:

  

Concept used:

If there is highlight dot in the graph means the limit exist on the point and if there is a hole in the graph at the value that x is approaching, with no other point for a different value of the function, then the limit does still exist.

Calculation:

A hole would be limit.

As holes approaches to the $y=3$ .

Therefore,

  $\underset{x\to 2}{\lim }f\left(x\right)=3$ .

Hence, $\underset{x\to 2}{\lim }f\left(x\right)=3$ .