Calculus Examples

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

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

Step 1

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

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

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

Step 2

Evaluate the limit of

x

$x$ by plugging in

0

$0$ for

x

$x$ .

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

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite

\tan (x)

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

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

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

Step 3.2

Multiply by the reciprocal of the fraction to divide by

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

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

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

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

Step 3.3

Write

\sin (0)

$\sin (0)$ as a fraction with denominator

1

$1$ .

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

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

Step 3.4

Simplify.

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Step 3.4.1

Divide

\sin (0)

$\sin (0)$ by

1

$1$ .

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

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

Step 3.4.2

Combine

\sin (0)

$\sin (0)$ and

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

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

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

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

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

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

Step 3.5

The exact value of

\sin (0)

$\sin (0)$ is

0

$0$ .

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

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

Step 3.6

Multiply

0

$0$ by

\cos (x)

$\cos (x)$ .

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

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

Step 3.7

Divide

0

$0$ by

\sin (x)

$\sin (x)$ .

0

$0$

0

$0$