Question

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

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

Short answer

The limit of function is$-1$.

Step-by-step solution

Step 1. Write the given expression.

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

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.

$|x+4|=\left\{\begin{array}{l}x+4\text{if}x+4\ge 0,x\ge -4\\ -(x+4)\text{if}x+4\le 0,x<-4\end{array}\right.$

Step 3. Find the limit of the function.

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

$\begin{array}{c}\underset{x\to -4^{-}}{\lim }\frac{|x+4|}{x+4}=\underset{x\to -4^{-}}{\lim }\frac{-(x+4)}{x+4}\\ =\underset{x\to -4^{-}}{\lim }(-1)\\ =-1\end{array}$

The limit exists and value of limit is $-1$.

Therefore, the limit of function is $-1$.