Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

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

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

Step 3.4.2

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

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