Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Evaluate the limit of $\frac{1}{2}$ which is constant as $x$ approaches $0$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $\frac{π}{6}$ and $0$ .

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

Step 3.1.2

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

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

Step 3.1.3

Combine the numerators over the common denominator.

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

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{\frac{0}{2}}{x}$

Step 3.1.5

Divide $0$ by $2$ .

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$