Question

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

$\underset{x\to 2}{\lim }\frac{|x-2|}{x-2}$

Short answer

The limitdoes not exist.

Step-by-step solution

Step 1. Write the given expression.

The expression is$\underset{x\to 2}{\lim }\frac{|x-2|}{x-2}$

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$2$from the right and one from the left.

$|x-2|=\left\{\begin{array}{l}x-2\text{if}x-2\ge 0,x\ge 2\\ -(x-2)\text{if}x-2\le 0,x<2\end{array}\right.$

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

As $x$approaches $2$ from the right,

$\begin{array}{c}\underset{x\to 2^{+}}{\lim }\frac{|x-2|}{x-2}=\underset{x\to 2^{+}}{\lim }\frac{x-2}{x-2}\\ =\underset{x\to 2^{+}}{\lim }1\\ =1\end{array}$

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

As $x$approaches $2$ from the left,

$\begin{array}{c}\underset{x\to 2^{-}}{\lim }\frac{|x-2|}{x-2}=\underset{x\to 2^{-}}{\lim }\frac{-(x-2)}{x-2}\\ =\underset{x\to 2^{-}}{\lim }-1\\ =-1\end{array}$

Since the limits of the absolute value functions as$x$approaches$2$from both sides are not equal, the limit does not exist.

Therefore, the limit of function does not exist.