Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Separate fractions.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify.

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Step 3.4.1

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

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

Step 3.4.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 3.5

Multiply $2$ by $0$ .

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

Step 3.6

Rewrite using the commutative property of multiplication.

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

Step 3.7

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

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

Step 3.8

Multiply $0$ by $\frac{1}{x}$ .

$0 \sec (x)$

Step 3.9

Multiply $0$ by $\sec (x)$ .

$0$