Question

Does the limit exist? Find the limit, if it exists. If the limit does not exist, explain why?

$\underset{x\to -4}{\lim }|x+4|$

Short answer

The limit of function is$0.$

Step-by-step solution

Step 1. Write the given expression.

The expression is$\underset{x\to -4}{\lim }|x+4|$

Step 2. Re-write the given function.

The given is the limit of an absolute value function, so one-sided limits must be used.

The absolute value function can be written as a piecewise function defined over two intervals, one wherein $x$approaches$-4$from the right and one from the left.

$|x+4|=\left\{\begin{array}{l}x+4\text{if}x\ge -4\\ -(x+4)\text{if}x<-4\end{array}\right.$

Step 3. Find the limit of the function from the right.

As $x$approaches $-4$ from the right,

$\begin{array}{c}\underset{x\to -4^{+}}{\lim }x+4=-4+4\\ =0\end{array}$

Step 4. Find the limit of the function from the left.

As $x$approaches $-4$ from the left,

$\begin{array}{c}\underset{x\to -4^{-}}{\lim }x+4=-(-4+4)\\ =0\end{array}$

Since the limits of the absolute value functions as $x$ approaches $-4$from both sides are equal, the limit exists.

Therefore, the limit of function is $0$.