Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $0$ and $\frac{π}{6}$ .

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

Step 3.1.2

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

$\frac{\frac{1}{2}}{x + \frac{π}{6}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{\frac{1}{2}}{x \cdot \frac{6}{6} + \frac{π}{6}}$

Step 3.2.2

Combine $x$ and $\frac{6}{6}$ .

$\frac{\frac{1}{2}}{\frac{x \cdot 6}{6} + \frac{π}{6}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{\frac{1}{2}}{\frac{x \cdot 6 + π}{6}}$

Step 3.2.4

Move $6$ to the left of $x$ .

$\frac{\frac{1}{2}}{\frac{6 x + π}{6}}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{6}{6 x + π}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Factor $2$ out of $6$ .

$\frac{1}{2} \cdot \frac{2 (3)}{6 x + π}$

Step 3.4.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 3}{6 x + π}$

Step 3.4.3

Rewrite the expression.

$\frac{3}{6 x + π}$