Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{6}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limits by plugging in $\frac{π}{6}$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

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

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

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

Step 5

Simplify each term.

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Step 5.1

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

$\frac{1}{2} + \tan (\frac{π}{6})$

Step 5.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{1}{2} + \frac{\sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} + \frac{\sqrt{3}}{3}$

Decimal Form:

$1.07735026 \dots$