Calculus Examples

$\frac{lim_{x \to 0} \frac{\sin (\frac{π}{6} + x) \cdot 1}{2}}{x}$

Step 1

Evaluate the limit.

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Step 1.1

Multiply $\sin (\frac{π}{6} + x)$ by $1$ .

$\frac{lim_{x \to 0} \frac{\sin (\frac{π}{6} + x)}{2}}{x}$

Step 1.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{1}{2} lim_{x \to 0} \sin (\frac{π}{6} + x)}{x}$

Step 1.3

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

$\frac{\frac{1}{2} \sin (lim_{x \to 0} \frac{π}{6} + x)}{x}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{2} \sin (lim_{x \to 0} \frac{π}{6} + lim_{x \to 0} x)}{x}$

Step 1.5

Evaluate the limit of $\frac{π}{6}$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{2} \sin (\frac{π}{6} + lim_{x \to 0} x)}{x}$

Step 2

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

$\frac{\frac{1}{2} \sin (\frac{π}{6} + 0)}{x}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{2}$ and $\sin (\frac{π}{6} + 0)$ .

$\frac{\frac{\sin (\frac{π}{6} + 0)}{2}}{x}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Add $\frac{π}{6}$ and $0$ .

$\frac{\frac{\sin (\frac{π}{6})}{2}}{x}$

Step 3.2.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{\frac{\frac{1}{2}}{2}}{x}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{1}{2} \cdot \frac{1}{2}}{x}$

Step 3.3.2

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{\frac{1}{2 \cdot 2}}{x}$

Step 3.3.2.2

Multiply $2$ by $2$ .

$\frac{\frac{1}{4}}{x}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{4} \cdot \frac{1}{x}$

Step 3.5

Multiply $\frac{1}{4}$ by $\frac{1}{x}$ .

$\frac{1}{4 x}$