Calculus Examples

$lim_{x \to - \frac{π}{6}} 4 \cos (x) - 3 \tan (- 2 x)$

Step 1

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

$lim_{x \to - \frac{π}{6}} 4 \cos (x) - lim_{x \to - \frac{π}{6}} 3 \tan (- 2 x)$

Step 2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to - \frac{π}{6}} \cos (x) - lim_{x \to - \frac{π}{6}} 3 \tan (- 2 x)$

Step 3

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

$4 \cos (lim_{x \to - \frac{π}{6}} x) - lim_{x \to - \frac{π}{6}} 3 \tan (- 2 x)$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$4 \cos (lim_{x \to - \frac{π}{6}} x) - 3 lim_{x \to - \frac{π}{6}} \tan (- 2 x)$

Step 5

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

$4 \cos (lim_{x \to - \frac{π}{6}} x) - 3 \tan (lim_{x \to - \frac{π}{6}} - 2 x)$

Step 6

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

$4 \cos (lim_{x \to - \frac{π}{6}} x) - 3 \tan (- 2 lim_{x \to - \frac{π}{6}} x)$

Step 7

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

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Step 7.1

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

$4 \cos (- \frac{π}{6}) - 3 \tan (- 2 lim_{x \to - \frac{π}{6}} x)$

Step 7.2

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

$4 \cos (- \frac{π}{6}) - 3 \tan (- 2 (- \frac{π}{6}))$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

$4 \cos (\frac{11 π}{6}) - 3 \tan (- 2 (- \frac{π}{6}))$

Step 8.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$4 \cos (\frac{π}{6}) - 3 \tan (- 2 (- \frac{π}{6}))$

Step 8.1.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$4 \frac{\sqrt{3}}{2} - 3 \tan (- 2 (- \frac{π}{6}))$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Factor $2$ out of $4$ .

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

Step 8.1.4.2

Cancel the common factor.

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

Step 8.1.4.3

Rewrite the expression.

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

Step 8.1.5

Cancel the common factor of $2$ .

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Step 8.1.5.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

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

Step 8.1.5.2

Factor $2$ out of $- 2$ .

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

Step 8.1.5.3

Factor $2$ out of $6$ .

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

Step 8.1.5.4

Cancel the common factor.

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

Step 8.1.5.5

Rewrite the expression.

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

Step 8.1.6

Move the negative in front of the fraction.

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

Step 8.1.7

Multiply $- - \frac{π}{3}$ .

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Step 8.1.7.1

Multiply $- 1$ by $- 1$ .

$2 \sqrt{3} - 3 \tan (1 \frac{π}{3})$

Step 8.1.7.2

Multiply $\frac{π}{3}$ by $1$ .

$2 \sqrt{3} - 3 \tan (\frac{π}{3})$

Step 8.1.8

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

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

Step 8.2

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

$- \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$