Calculus Examples

$lim_{x \to 0} \frac{\sin (x)}{| x |}$

Step 1

Consider the left sided limit.

$lim_{x \to 0^{-}} \frac{\sin (x)}{| x |}$

Step 2

Make a table to show the behavior of the function $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \frac{\sin (x)}{| x |} \\ - 0.1 & - 0.99833416 \\ - 0.01 & - 0.99998333 \\ - 0.001 & - 0.99999983 \end{array}$

Step 3

As the $x$ values approach $0$ , the function values approach $- 1$ . Thus, the limit of $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the left is $- 1$ .

$- 1$

Step 4

Consider the right sided limit.

$lim_{x \to 0^{+}} \frac{\sin (x)}{| x |}$

Step 5

Make a table to show the behavior of the function $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \frac{\sin (x)}{| x |} \\ 0.1 & 0.99833416 \\ 0.01 & 0.99998333 \\ 0.001 & 0.99999983 \end{array}$

Step 6

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the right is $1$ .

$1$

Step 7

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

$Does not exist$