Precalculus Examples

$lim_{x \to 0} \frac{x^{2} + 2 | x |}{x}$

Step 1

Consider the left sided limit.

$lim_{x \to 0^{-}} \frac{x^{2} + 2 | x |}{x}$

Step 2

Make a table to show the behavior of the function $\frac{x^{2} + 2 | x |}{x}$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \frac{x^{2} + 2 | x |}{x} \\ - 0.1 & - 2.1 \\ - 0.01 & - 2.01 \\ - 0.001 & - 2.001 \\ - 0.0001 & - 2.0001 \\ - 0.00001 & - 2.00001 \end{array}$

Step 3

As the $x$ values approach $0$ , the function values approach $- 2$ . Thus, the limit of $\frac{x^{2} + 2 | x |}{x}$ as $x$ approaches $0$ from the left is $- 2$ .

$- 2$

Step 4

Consider the right sided limit.

$lim_{x \to 0^{+}} \frac{x^{2} + 2 | x |}{x}$

Step 5

Make a table to show the behavior of the function $\frac{x^{2} + 2 | x |}{x}$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \frac{x^{2} + 2 | x |}{x} \\ 0.1 & 2.1 \\ 0.01 & 2.01 \\ 0.001 & 2.001 \\ 0.0001 & 2.0001 \\ 0.00001 & 2.00001 \end{array}$

Step 6

As the $x$ values approach $0$ , the function values approach $2$ . Thus, the limit of $\frac{x^{2} + 2 | x |}{x}$ as $x$ approaches $0$ from the right is $2$ .

$2$

Step 7

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

$Does not exist$