Precalculus Examples

$lim_{x \to 0} \frac{\cot (2 x)}{\csc (6 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\csc (6 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\cot (2 x)}{\frac{1}{\sin (6 x)}}$

Step 1.2

Rewrite $\cot (2 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\cos (2 x)}{\sin (2 x)}}{\frac{1}{\sin (6 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (6 x)}$ .

$lim_{x \to 0} \frac{\cos (2 x)}{\sin (2 x)} \sin (6 x)$

Step 1.4

Write $\sin (6 x)$ as a fraction with denominator $1$ .

$lim_{x \to 0} \frac{\cos (2 x)}{\sin (2 x)} \cdot \frac{\sin (6 x)}{1}$

Step 1.5

Simplify.

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Step 1.5.1

Divide $\sin (6 x)$ by $1$ .

$lim_{x \to 0} \frac{\cos (2 x)}{\sin (2 x)} \sin (6 x)$

Step 1.5.2

Convert from $\frac{\cos (2 x)}{\sin (2 x)}$ to $\cot (2 x)$ .

$lim_{x \to 0} \cot (2 x) \sin (6 x)$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \cot (2 x) \sin (6 x)$

Step 3

Make a table to show the behavior of the function $\cot (2 x) \sin (6 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \cot (2 x) \sin (6 x) \\ - 0.1 & 2.78546877 \\ - 0.01 & 2.99780055 \\ - 0.001 & 2.999978 \\ - 0.0001 & 2.99999978 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $3$ . Thus, the limit of $\cot (2 x) \sin (6 x)$ as $x$ approaches $0$ from the left is $3$ .

$3$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \cot (2 x) \sin (6 x)$

Step 6

Make a table to show the behavior of the function $\cot (2 x) \sin (6 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \cot (2 x) \sin (6 x) \\ 0.1 & 2.78546877 \\ 0.01 & 2.99780055 \\ 0.001 & 2.999978 \\ 0.0001 & 2.99999978 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $3$ . Thus, the limit of $\cot (2 x) \sin (6 x)$ as $x$ approaches $0$ from the right is $3$ .

$3$