Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Rewrite $\frac{\frac{\sin (0)}{\cos (0)}}{\sin (x)}$ as a product.

$\frac{\sin (0)}{\cos (0)} \cdot \frac{1}{\sin (x)}$

Step 3.3

Multiply $\frac{\sin (0)}{\cos (0)}$ by $\frac{1}{\sin (x)}$ .

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

Step 3.4

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

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

The exact value of $\cos (0)$ is $1$ .

$\frac{0}{1 \sin (x)}$

Step 3.5.2

Multiply $\sin (x)$ by $1$ .

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

Step 3.6

Divide $0$ by $\sin (x)$ .

$0$