Calculus Examples

$lim_{t \to 0} \tan (1 - \frac{\sin (t)}{t})$

Step 1

Consider the left sided limit.

$lim_{t \to 0^{-}} \tan (1 - \frac{\sin (t)}{t})$

Step 2

Make a table to show the behavior of the function $\tan (1 - \frac{\sin (t)}{t})$ as $t$ approaches $0$ from the left.

$\begin{array}{c} t & \tan (1 - \frac{\sin (t)}{t}) \\ - 0.1 & 0.00166583 \\ - 0.01 & 0.00001666 \\ - 0.001 & 0.00000016 \end{array}$

Step 3

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

$0$

Step 4

Consider the right sided limit.

$lim_{t \to 0^{+}} \tan (1 - \frac{\sin (t)}{t})$

Step 5

Make a table to show the behavior of the function $\tan (1 - \frac{\sin (t)}{t})$ as $t$ approaches $0$ from the right.

$\begin{array}{c} t & \tan (1 - \frac{\sin (t)}{t}) \\ 0.1 & 0.00166583 \\ 0.01 & 0.00001666 \\ 0.001 & 0.00000016 \end{array}$

Step 6

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

$0$