Chapter 11, Problem 1RE

To determine

The value of $\underset{x\to -3}{\lim }f\left(x\right);\underset{x\to -3^{-}}{\lim }f\left(x\right)\underset{x\to -3^{+}}{;\lim }f\left(x\right)$.

Answer to Problem 1RE

Solution:

$\underset{\_}{\underset{x\to -3}{\lim }f\left(x\right)=\underset{x\to -3^{{}^{-}}}{\lim }f\left(x\right)=\underset{x\to -3^{+}}{\lim }f\left(x\right)=4}$

Explanation of Solution

Given:

Graph of $f\left(x\right)$ is shown below:

Explanation:

The graph shows that as $x$ approaches -3 from the left, the corresponding values of $f\left(x\right)$ approach 4.

Therefore,

$\underset{x\to -3^{{}^{-}}}{\lim }f\left(x\right)=4$

The graph shows that as $x$ approaches -3 from the right, the corresponding values of $f\left(x\right)$ approach 4.

Therefore,

$\underset{x\to -3^{{}^{+}}}{\lim }f\left(x\right)=4$

Since the left-hand and right-hand limit exists and are equal.

Therefore,

$\underset{x\to -3}{\lim }f\left(x\right)$ exists and is equal to 4.