Calculus Examples

$lim_{x \to 2} \frac{| x - 2 |}{x - 2}$

Step 1

Consider the left sided limit.

$lim_{x \to 2^{-}} \frac{| x - 2 |}{x - 2}$

Step 2

Make a table to show the behavior of the function $\frac{| x - 2 |}{x - 2}$ as $x$ approaches $2$ from the left.

$\begin{array}{c} x & \frac{| x - 2 |}{x - 2} \\ 1.9 & - 1 \\ 1.99 & - 1 \\ 1.999 & - 1 \end{array}$

Step 3

As the $x$ values approach $2$ , the function values approach $- 1$ . Thus, the limit of $\frac{| x - 2 |}{x - 2}$ as $x$ approaches $2$ from the left is $- 1$ .

$- 1$

Step 4

Consider the right sided limit.

$lim_{x \to 2^{+}} \frac{| x - 2 |}{x - 2}$

Step 5

Make a table to show the behavior of the function $\frac{| x - 2 |}{x - 2}$ as $x$ approaches $2$ from the right.

$\begin{array}{c} x & \frac{| x - 2 |}{x - 2} \\ 2.1 & 1 \\ 2.01 & 1 \\ 2.001 & 1 \end{array}$

Step 6

As the $x$ values approach $2$ , the function values approach $1$ . Thus, the limit of $\frac{| x - 2 |}{x - 2}$ as $x$ approaches $2$ from the right is $1$ .

$1$

Step 7

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$