Calculus Examples

$lim_{x \to - 5} \frac{| x + 5 |}{x + 5}$

Step 1

Consider the left sided limit.

$lim_{x \to {(- 5)}^{-}} \frac{| x + 5 |}{x + 5}$

Step 2

Make a table to show the behavior of the function $\frac{| x + 5 |}{x + 5}$ as $x$ approaches $- 5$ from the left.

$\begin{array}{c} x & \frac{| x + 5 |}{x + 5} \\ - 5.1 & - 1 \\ - 5.01 & - 1 \\ - 5.001 & - 1 \end{array}$

Step 3

As the $x$ values approach $- 5$ , the function values approach $- 1$ . Thus, the limit of $\frac{| x + 5 |}{x + 5}$ as $x$ approaches $- 5$ from the left is $- 1$ .

$- 1$

Step 4

Consider the right sided limit.

$lim_{x \to {(- 5)}^{+}} \frac{| x + 5 |}{x + 5}$

Step 5

Make a table to show the behavior of the function $\frac{| x + 5 |}{x + 5}$ as $x$ approaches $- 5$ from the right.

$\begin{array}{c} x & \frac{| x + 5 |}{x + 5} \\ - 4.9 & 1 \\ - 4.99 & 1 \\ - 4.999 & 1 \end{array}$

Step 6

As the $x$ values approach $- 5$ , the function values approach $1$ . Thus, the limit of $\frac{| x + 5 |}{x + 5}$ as $x$ approaches $- 5$ from the right is $1$ .

$1$

Step 7

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

$Does not exist$