Calculus Examples

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

Step 1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$lim_{x \to 0} \csc (x) - \frac{1}{x}$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \csc (x) - \frac{1}{x}$

Step 3

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

$\begin{array}{c} x & \csc (x) - \frac{1}{x} \\ - 0.1 & - 0.01668613 \\ - 0.01 & - 0.00166668 \\ - 0.001 & - 0.00016666 \\ - 0.0001 & - 0.00001666 \end{array}$

Step 4

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

$0$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \csc (x) - \frac{1}{x}$

Step 6

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

$\begin{array}{c} x & \csc (x) - \frac{1}{x} \\ 0.1 & 0.01668613 \\ 0.01 & 0.00166668 \\ 0.001 & 0.00016666 \\ 0.0001 & 0.00001666 \end{array}$

Step 7

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

$0$