Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite $\tan (2 \cdot 0)$ in terms of sines and cosines.

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

Step 3.2

Rewrite $\frac{\frac{\sin (2 \cdot 0)}{\cos (2 \cdot 0)}}{\sin (5 x)}$ as a product.

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $0$ .

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

Step 3.5

Multiply $2$ by $0$ .

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

Step 3.6

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

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

Step 3.7

Simplify the denominator.

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Step 3.7.1

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

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

Step 3.7.2

Multiply $\sin (5 x)$ by $1$ .

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

Step 3.8

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

$0$