Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.1.5

Move the negative in front of the fraction.

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

Step 9.1.6

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

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

Step 9.1.7

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.

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

Step 9.1.8

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

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

Step 9.1.9

Multiply $- 4 (- \frac{\sqrt{3}}{3})$ .

Tap for more steps...

Step 9.1.9.1

Multiply $- 1$ by $- 4$ .

$- \frac{3 \sqrt{3}}{2} + 4 \frac{\sqrt{3}}{3}$

Step 9.1.9.2

Combine $4$ and $\frac{\sqrt{3}}{3}$ .

$- \frac{3 \sqrt{3}}{2} + \frac{4 \sqrt{3}}{3}$

Step 9.2

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

$- \frac{3 \sqrt{3}}{2} \cdot \frac{3}{3} + \frac{4 \sqrt{3}}{3}$

Step 9.3

To write $\frac{4 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 \sqrt{3}}{2} \cdot \frac{3}{3} + \frac{4 \sqrt{3}}{3} \cdot \frac{2}{2}$

Step 9.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.4.1

Multiply $\frac{3 \sqrt{3}}{2}$ by $\frac{3}{3}$ .

$- \frac{3 \sqrt{3} \cdot 3}{2 \cdot 3} + \frac{4 \sqrt{3}}{3} \cdot \frac{2}{2}$

Step 9.4.2

Multiply $2$ by $3$ .

$- \frac{3 \sqrt{3} \cdot 3}{6} + \frac{4 \sqrt{3}}{3} \cdot \frac{2}{2}$

Step 9.4.3

Multiply $\frac{4 \sqrt{3}}{3}$ by $\frac{2}{2}$ .

$- \frac{3 \sqrt{3} \cdot 3}{6} + \frac{4 \sqrt{3} \cdot 2}{3 \cdot 2}$

Step 9.4.4

Multiply $3$ by $2$ .

$- \frac{3 \sqrt{3} \cdot 3}{6} + \frac{4 \sqrt{3} \cdot 2}{6}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{- 3 \sqrt{3} \cdot 3 + 4 \sqrt{3} \cdot 2}{6}$

Step 9.6

Simplify the numerator.

Tap for more steps...

Step 9.6.1

Multiply $3$ by $- 3$ .

$\frac{- 9 \sqrt{3} + 4 \sqrt{3} \cdot 2}{6}$

Step 9.6.2

Multiply $2$ by $4$ .

$\frac{- 9 \sqrt{3} + 8 \sqrt{3}}{6}$

Step 9.6.3

Add $- 9 \sqrt{3}$ and $8 \sqrt{3}$ .

$\frac{- \sqrt{3}}{6}$

Step 9.7

Move the negative in front of the fraction.

$- \frac{\sqrt{3}}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3}}{6}$

Decimal Form:

$- 0.28867513 \dots$