Calculus Examples

$\frac{lim_{x \to 0} \sin (2 x)}{\tan (5 x)}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 0} 2 x)}{\tan (5 x)}$

Step 1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (2 lim_{x \to 0} x)}{\tan (5 x)}$

Step 2

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

$\frac{\sin (2 \cdot 0)}{\tan (5 x)}$

Step 3

Simplify the answer.

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Step 3.1

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

$\frac{\sin (2 \cdot 0)}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 3.2

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

$\sin (2 \cdot 0) \frac{\cos (5 x)}{\sin (5 x)}$

Step 3.3

Write $\sin (2 \cdot 0)$ as a fraction with denominator $1$ .

$\frac{\sin (2 \cdot 0)}{1} \cdot \frac{\cos (5 x)}{\sin (5 x)}$

Step 3.4

Simplify.

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Step 3.4.1

Divide $\sin (2 \cdot 0)$ by $1$ .

$\sin (2 \cdot 0) \frac{\cos (5 x)}{\sin (5 x)}$

Step 3.4.2

Combine $\sin (2 \cdot 0)$ and $\frac{\cos (5 x)}{\sin (5 x)}$ .

$\frac{\sin (2 \cdot 0) \cos (5 x)}{\sin (5 x)}$

Step 3.5

Multiply $2$ by $0$ .

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

Step 3.6

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

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

Step 3.7

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

$\frac{0}{\sin (5 x)}$

Step 3.8

Divide $0$ by $\sin (5 x)$ .

$0$