Calculus Examples

$lim_{x \to - \frac{π}{3}} \frac{- 4 \sin (x)}{- 8 + 5 \cos (x)}$

Step 1

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

$- 4 lim_{x \to - \frac{π}{3}} \frac{\sin (x)}{- 8 + 5 \cos (x)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- \frac{π}{3}$ .

$- 4 \frac{lim_{x \to - \frac{π}{3}} \sin (x)}{lim_{x \to - \frac{π}{3}} - 8 + 5 \cos (x)}$

Step 3

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

$- 4 \frac{\sin (lim_{x \to - \frac{π}{3}} x)}{lim_{x \to - \frac{π}{3}} - 8 + 5 \cos (x)}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{π}{3}$ .

$- 4 \frac{\sin (lim_{x \to - \frac{π}{3}} x)}{- lim_{x \to - \frac{π}{3}} 8 + lim_{x \to - \frac{π}{3}} 5 \cos (x)}$

Step 5

Evaluate the limit of $8$ which is constant as $x$ approaches $- \frac{π}{3}$ .

$- 4 \frac{\sin (lim_{x \to - \frac{π}{3}} x)}{- 1 \cdot 8 + lim_{x \to - \frac{π}{3}} 5 \cos (x)}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$- 4 \frac{\sin (lim_{x \to - \frac{π}{3}} x)}{- 8 + 5 lim_{x \to - \frac{π}{3}} \cos (x)}$

Step 7

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

$- 4 \frac{\sin (lim_{x \to - \frac{π}{3}} x)}{- 8 + 5 \cos (lim_{x \to - \frac{π}{3}} x)}$

Step 8

Evaluate the limits by plugging in $- \frac{π}{3}$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- \frac{π}{3}$ for $x$ .

$- 4 \frac{\sin (- \frac{π}{3})}{- 8 + 5 \cos (lim_{x \to - \frac{π}{3}} x)}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- \frac{π}{3}$ for $x$ .

$- 4 \frac{\sin (- \frac{π}{3})}{- 8 + 5 \cos (- \frac{π}{3})}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$- 4 \frac{\sin (\frac{5 π}{3})}{- 8 + 5 \cos (- \frac{π}{3})}$

Step 9.1.2

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

$- 4 \frac{- \sin (\frac{π}{3})}{- 8 + 5 \cos (- \frac{π}{3})}$

Step 9.1.3

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

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 + 5 \cos (- \frac{π}{3})}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 + 5 \cos (\frac{5 π}{3})}$

Step 9.2.2

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

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 + 5 \cos (\frac{π}{3})}$

Step 9.2.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 + 5 (\frac{1}{2})}$

Step 9.2.4

Combine $5$ and $\frac{1}{2}$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 + \frac{5}{2}}$

Step 9.2.5

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- 8 \cdot \frac{2}{2} + \frac{5}{2}}$

Step 9.2.6

Combine $- 8$ and $\frac{2}{2}$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{\frac{- 8 \cdot 2}{2} + \frac{5}{2}}$

Step 9.2.7

Combine the numerators over the common denominator.

$- 4 \frac{- \frac{\sqrt{3}}{2}}{\frac{- 8 \cdot 2 + 5}{2}}$

Step 9.2.8

Simplify the numerator.

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Step 9.2.8.1

Multiply $- 8$ by $2$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{\frac{- 16 + 5}{2}}$

Step 9.2.8.2

Add $- 16$ and $5$ .

$- 4 \frac{- \frac{\sqrt{3}}{2}}{\frac{- 11}{2}}$

Step 9.2.9

Move the negative in front of the fraction.

$- 4 \frac{- \frac{\sqrt{3}}{2}}{- \frac{11}{2}}$

Step 9.3

Dividing two negative values results in a positive value.

$- 4 \frac{\frac{\sqrt{3}}{2}}{\frac{11}{2}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$- 4 (\frac{\sqrt{3}}{2} \cdot \frac{2}{11})$

Step 9.5

Cancel the common factor of $2$ .

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Step 9.5.1

Cancel the common factor.

$- 4 (\frac{\sqrt{3}}{2} \cdot \frac{2}{11})$

Step 9.5.2

Rewrite the expression.

$- 4 (\sqrt{3} \frac{1}{11})$

Step 9.6

Combine $\sqrt{3}$ and $\frac{1}{11}$ .

$- 4 \frac{\sqrt{3}}{11}$

Step 9.7

Combine $- 4$ and $\frac{\sqrt{3}}{11}$ .

$\frac{- 4 \sqrt{3}}{11}$

Step 9.8

Move the negative in front of the fraction.

$- \frac{4 \sqrt{3}}{11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 \sqrt{3}}{11}$

Decimal Form:

$- 0.62983665 \dots$