Precalculus Examples

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

Step 1

Consider the left sided limit.

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

Step 2

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

$\begin{array}{c} x & \csc (x) - \cot (x) \\ - 0.1 & - 0.0500417 \\ - 0.01 & - 0.00500004 \\ - 0.001 & - 0.0005 \\ - 0.0001 & - 0.00005 \\ - 0.00001 & - 0.000005 \end{array}$

Step 3

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

$0$

Step 4

Consider the right sided limit.

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

Step 5

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

$\begin{array}{c} x & \csc (x) - \cot (x) \\ 0.1 & 0.0500417 \\ 0.01 & 0.00500004 \\ 0.001 & 0.0005 \\ 0.0001 & 0.00005 \\ 0.00001 & 0.000005 \end{array}$

Step 6

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

$\frac{0}{(x) \csc (x)}$

Step 7

Simplify the answer.

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Step 7.1

Separate fractions.

$\frac{0}{x} \cdot \frac{1}{\csc (x)}$

Step 7.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{0}{x} \cdot \frac{1}{\frac{1}{\sin (x)}}$

Step 7.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\frac{0}{x} (1 \sin (x))$

Step 7.4

Multiply $\sin (x)$ by $1$ .

$\frac{0}{x} \sin (x)$

Step 7.5

Divide $0$ by $x$ .

$0 \sin (x)$

Step 7.6

Multiply $0$ by $\sin (x)$ .

$0$