Calculus Examples

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

Step 1

Consider the left sided limit.

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

Step 2

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

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

Step 3

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

$1$

Step 4

Consider the right sided limit.

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

Step 5

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

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

Step 6

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

$Does not exist$