Calculus Examples

$\frac{lim_{x \to 0} \sin (6 x)}{(x) \sec (6 x)}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 0} 6 x)}{(x) \sec (6 x)}$

Step 1.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (6 lim_{x \to 0} x)}{(x) \sec (6 x)}$

Step 2

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

$\frac{\sin (6 \cdot 0)}{(x) \sec (6 x)}$

Step 3

Simplify the answer.

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Step 3.1

Separate fractions.

$\frac{1}{x} \cdot \frac{\sin (6 \cdot 0)}{\sec (6 x)}$

Step 3.2

Rewrite $\sec (6 x)$ in terms of sines and cosines.

$\frac{1}{x} \cdot \frac{\sin (6 \cdot 0)}{\frac{1}{\cos (6 x)}}$

Step 3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (6 x)}$ .

$\frac{1}{x} (\sin (6 \cdot 0) \cos (6 x))$

Step 3.4

Multiply $6$ by $0$ .

$\frac{1}{x} (\sin (0) \cos (6 x))$

Step 3.5

Rewrite using the commutative property of multiplication.

$\sin (0) \frac{1}{x} \cos (6 x)$

Step 3.6

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

$0 \frac{1}{x} \cos (6 x)$

Step 3.7

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

$0 \cos (6 x)$

Step 3.8

Multiply $0$ by $\cos (6 x)$ .

$0$