Textbook Problem

In Problems 1-6, use the graph of $y=f\left(x\right)$ in Figure 9.40 to find the functional values and limits, if they exist.

$1.(a)f(-2)(b)\underset{x\to -2}{\lim }f(x)$

To determine

The value of $f\left(-2\right)$ by using the provided graph.

Answer to Problem 1RE

Solution:

The value of $f\left(-2\right)$ is 2.

Explanation of Solution

Given Information:

The graph of $y=f\left(x\right)$ is shown below,

The function is $f\left(-2\right)$.

Explanation:

Consider the provided graph,

The value of c is equal to $-2$, so, the value of $f\left(-2\right)$ is equal to the y-coordinate at $x=-2$, which is 2.

Hence, the value of $f\left(-2\right)$ is 2.

To determine

The value of $\underset{x\to -2}{\lim }f\left(x\right)$ by using the provided graph.

Answer to Problem 1RE

Solution:

The value of $\underset{x\to -2}{\lim }f\left(x\right)$ is 2.

Explanation of Solution

Given Information:

The graph of $y=f\left(x\right)$ is shown below,

The value of c is $-2$.

Explanation:

Consider the provided graph,

The value of c is equal to $-2$.

The limit from the left is represented by $\underset{x\to -2^{-}}{\lim }f\left(x\right)$ and the limit from the right is represented by $\underset{x\to -2^{+}}{\lim }f\left(x\right)$.

The $\underset{x\to c}{\lim }f\left(x\right)$ will exist at $c=-2$ when the limit from the left, that is, the values of $f\left(c\right)$ but $c<-2$, is equal to the limit from the right, that is, the values of $f\left(c\right)$ but $c>-2$.

From the graph,

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

And,

$\underset{x\to -2^{+}}{\lim }f\left(x\right)=2$

Since,

$\underset{x\to -2^{-}}{\lim }f\left(x\right)=\underset{x\to -2^{+}}{\lim }f\left(x\right)$

Therefore, the value of $\underset{x\to -2}{\lim }f\left(x\right)$ is 2.