Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$- 4 \frac{\cos (\frac{5 π}{3})}{3 - 2 \sin (- 2 \frac{π}{3})}$

Step 11.1.2

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

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

Step 11.1.3

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

$- 4 \frac{\frac{1}{2}}{3 - 2 \sin (- 2 \frac{π}{3})}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Combine $- 2$ and $\frac{π}{3}$ .

$- 4 \frac{\frac{1}{2}}{3 - 2 \sin (\frac{- 2 π}{3})}$

Step 11.2.2

Move the negative in front of the fraction.

$- 4 \frac{\frac{1}{2}}{3 - 2 \sin (- \frac{2 π}{3})}$

Step 11.2.3

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

$- 4 \frac{\frac{1}{2}}{3 - 2 \sin (\frac{4 π}{3})}$

Step 11.2.4

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 third quadrant.

$- 4 \frac{\frac{1}{2}}{3 - 2 (- \sin (\frac{π}{3}))}$

Step 11.2.5

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

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

Step 11.2.6

Cancel the common factor of $2$ .

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Step 11.2.6.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$- 4 \frac{\frac{1}{2}}{3 - 2 \frac{- \sqrt{3}}{2}}$

Step 11.2.6.2

Factor $2$ out of $- 2$ .

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

Step 11.2.6.3

Cancel the common factor.

$- 4 \frac{\frac{1}{2}}{3 + 2 \cdot - 1 \frac{- \sqrt{3}}{2}}$

Step 11.2.6.4

Rewrite the expression.

$- 4 \frac{\frac{1}{2}}{3 - - \sqrt{3}}$

Step 11.2.7

Multiply $- 1$ by $- 1$ .

$- 4 \frac{\frac{1}{2}}{3 + 1 \sqrt{3}}$

Step 11.2.8

Multiply $\sqrt{3}$ by $1$ .

$- 4 \frac{\frac{1}{2}}{3 + \sqrt{3}}$

Step 11.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.4

Multiply $\frac{1}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

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

Step 11.5

Multiply $\frac{1}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

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

Step 11.6

Expand the denominator using the FOIL method.

$- 4 (\frac{1}{2} \cdot \frac{3 - \sqrt{3}}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}})$

Step 11.7

Simplify.

$- 4 (\frac{1}{2} \cdot \frac{3 - \sqrt{3}}{6})$

Step 11.8

Multiply $\frac{1}{2} \cdot \frac{3 - \sqrt{3}}{6}$ .

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Step 11.8.1

Multiply $\frac{1}{2}$ by $\frac{3 - \sqrt{3}}{6}$ .

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

Step 11.8.2

Multiply $2$ by $6$ .

$- 4 \frac{3 - \sqrt{3}}{12}$

Step 11.9

Cancel the common factor of $4$ .

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Step 11.9.1

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{3 - \sqrt{3}}{12}$

Step 11.9.2

Factor $4$ out of $12$ .

$4 \cdot - 1 \frac{3 - \sqrt{3}}{4 \cdot 3}$

Step 11.9.3

Cancel the common factor.

$4 \cdot - 1 \frac{3 - \sqrt{3}}{4 \cdot 3}$

Step 11.9.4

Rewrite the expression.

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.42264973 \dots$