Calculus Examples

$lim_{x \to \frac{π}{6}} \tan (- 2 x) - \sin (2 x)$

Step 1

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

$lim_{x \to \frac{π}{6}} \tan (- 2 x) - lim_{x \to \frac{π}{6}} \sin (2 x)$

Step 2

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

$\tan (lim_{x \to \frac{π}{6}} - 2 x) - lim_{x \to \frac{π}{6}} \sin (2 x)$

Step 3

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

$\tan (- 2 lim_{x \to \frac{π}{6}} x) - lim_{x \to \frac{π}{6}} \sin (2 x)$

Step 4

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

$\tan (- 2 lim_{x \to \frac{π}{6}} x) - \sin (lim_{x \to \frac{π}{6}} 2 x)$

Step 5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\tan (- 2 lim_{x \to \frac{π}{6}} x) - \sin (2 lim_{x \to \frac{π}{6}} x)$

Step 6

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

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Step 6.1

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

$\tan (- 2 \frac{π}{6}) - \sin (2 lim_{x \to \frac{π}{6}} x)$

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Cancel the common factor of $2$ .

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Step 7.1.1.1

Factor $2$ out of $- 2$ .

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

Step 7.1.1.2

Factor $2$ out of $6$ .

$\tan (2 \cdot - 1 \frac{π}{2 \cdot 3}) - \sin (2 \frac{π}{6})$

Step 7.1.1.3

Cancel the common factor.

$\tan (2 \cdot - 1 \frac{π}{2 \cdot 3}) - \sin (2 \frac{π}{6})$

Step 7.1.1.4

Rewrite the expression.

$\tan (- \frac{π}{3}) - \sin (2 \frac{π}{6})$

Step 7.1.2

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\tan (\frac{5 π}{3}) - \sin (2 \frac{π}{6})$

Step 7.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$- \tan (\frac{π}{3}) - \sin (2 \frac{π}{6})$

Step 7.1.4

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

$- \sqrt{3} - \sin (2 \frac{π}{6})$

Step 7.1.5

Cancel the common factor of $2$ .

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Step 7.1.5.1

Factor $2$ out of $6$ .

$- \sqrt{3} - \sin (2 \frac{π}{2 (3)})$

Step 7.1.5.2

Cancel the common factor.

$- \sqrt{3} - \sin (2 \frac{π}{2 \cdot 3})$

Step 7.1.5.3

Rewrite the expression.

$- \sqrt{3} - \sin (\frac{π}{3})$

Step 7.1.6

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$- \sqrt{3} - \frac{\sqrt{3}}{2}$

Step 7.2

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \sqrt{3} \cdot \frac{2}{2} - \frac{\sqrt{3}}{2}$

Step 7.3

Combine $- \sqrt{3}$ and $\frac{2}{2}$ .

$\frac{- \sqrt{3} \cdot 2}{2} - \frac{\sqrt{3}}{2}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{- \sqrt{3} \cdot 2 - \sqrt{3}}{2}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Multiply $2$ by $- 1$ .

$\frac{- 2 \sqrt{3} - \sqrt{3}}{2}$

Step 7.5.2

Subtract $\sqrt{3}$ from $- 2 \sqrt{3}$ .

$\frac{- 3 \sqrt{3}}{2}$

Step 7.6

Move the negative in front of the fraction.

$- \frac{3 \sqrt{3}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 \sqrt{3}}{2}$

Decimal Form:

$- 2.59807621 \dots$