Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the trig function because tangent is continuous.

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

Step 1.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Rewrite $\tan (3 \cdot 0)$ in terms of sines and cosines.

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

Convert from $\frac{\sin (3 \cdot 0)}{\cos (3 \cdot 0)}$ to $\tan (3 \cdot 0)$ .

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

Step 3.4.2

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

$\tan (3 \cdot 0) \cot (5 x)$

Step 3.5

Multiply $3$ by $0$ .

$\tan (0) \cot (5 x)$

Step 3.6

The exact value of $\tan (0)$ is $0$ .

$0 \cot (5 x)$

Step 3.7

Multiply $0$ by $\cot (5 x)$ .

$0$