Calculus Examples

$\frac{lim_{x \to 0} \tan (x)}{\tan (2 x)}$

Step 1

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

$\frac{\tan (lim_{x \to 0} x)}{\tan (2 x)}$

Step 2

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

$\frac{\tan (0)}{\tan (2 x)}$

Step 3

Simplify the answer.

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Step 3.1

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

$\frac{\tan (0)}{\frac{\sin (2 x)}{\cos (2 x)}}$

Step 3.2

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

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

Step 3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (2 x)}{\cos (2 x)}$ .

$\frac{\sin (0)}{\cos (0)} \cdot \frac{\cos (2 x)}{\sin (2 x)}$

Step 3.4

Simplify.

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Step 3.4.1

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

$\tan (0) \frac{\cos (2 x)}{\sin (2 x)}$

Step 3.4.2

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

$\tan (0) \cot (2 x)$

Step 3.5

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

$0 \cot (2 x)$

Step 3.6

Multiply $0$ by $\cot (2 x)$ .

$0$