Calculus Examples

$lim_{x \to 0} \frac{\tan (3 x)}{\tan (5 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (5 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\tan (3 x)}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 1.2

Rewrite $\tan (3 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (3 x)}{\cos (3 x)}}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (5 x)}{\cos (5 x)}$ .

$lim_{x \to 0} \frac{\sin (3 x)}{\cos (3 x)} \cdot \frac{\cos (5 x)}{\sin (5 x)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{\sin (3 x)}{\cos (3 x)}$ to $\tan (3 x)$ .

$lim_{x \to 0} \tan (3 x) \frac{\cos (5 x)}{\sin (5 x)}$

Step 1.4.2

Convert from $\frac{\cos (5 x)}{\sin (5 x)}$ to $\cot (5 x)$ .

$lim_{x \to 0} \tan (3 x) \cot (5 x)$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \tan (3 x) \cot (5 x)$

Step 3

Make a table to show the behavior of the function $\tan (3 x) \cot (5 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \tan (3 x) \cot (5 x) \\ - 0.1 & 0.5662362 \\ - 0.01 & 0.59967983 \\ - 0.001 & 0.59999679 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $0.6$ . Thus, the limit of $\tan (3 x) \cot (5 x)$ as $x$ approaches $0$ from the left is $0.6$ .

$0.6$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} \tan (3 x) \cot (5 x)$

Step 6

Make a table to show the behavior of the function $\tan (3 x) \cot (5 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \tan (3 x) \cot (5 x) \\ 0.1 & 0.5662362 \\ 0.01 & 0.59967983 \\ 0.001 & 0.59999679 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $0.6$ . Thus, the limit of $\tan (3 x) \cot (5 x)$ as $x$ approaches $0$ from the right is $0.6$ .

$0.6$