Calculus Examples

$lim_{x \to \frac{π}{3}} \frac{2 \cos^{2} (x) + 3 \cos (x) - 2}{2 \cos (x) - 1}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \frac{π}{3}} 2 \cos^{2} (x) + 3 \cos (x) - 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to \frac{π}{3}} 2 \cos^{2} (x) + lim_{x \to \frac{π}{3}} 3 \cos (x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.2

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

$\frac{2 lim_{x \to \frac{π}{3}} \cos^{2} (x) + lim_{x \to \frac{π}{3}} 3 \cos (x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.3

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{2 {(lim_{x \to \frac{π}{3}} \cos (x))}^{2} + lim_{x \to \frac{π}{3}} 3 \cos (x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.4

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

$\frac{2 \cos^{2} (lim_{x \to \frac{π}{3}} x) + lim_{x \to \frac{π}{3}} 3 \cos (x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.5

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

$\frac{2 \cos^{2} (lim_{x \to \frac{π}{3}} x) + 3 lim_{x \to \frac{π}{3}} \cos (x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.6

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

$\frac{2 \cos^{2} (lim_{x \to \frac{π}{3}} x) + 3 \cos (lim_{x \to \frac{π}{3}} x) - lim_{x \to \frac{π}{3}} 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.7

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{π}{3}$ .

$\frac{2 \cos^{2} (lim_{x \to \frac{π}{3}} x) + 3 \cos (lim_{x \to \frac{π}{3}} x) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.8

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

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Step 1.1.2.8.1

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

$\frac{2 \cos^{2} (\frac{π}{3}) + 3 \cos (lim_{x \to \frac{π}{3}} x) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.8.2

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

$\frac{2 \cos^{2} (\frac{π}{3}) + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9

Simplify the answer.

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Step 1.1.2.9.1

Simplify each term.

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Step 1.1.2.9.1.1

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

$\frac{2 {(\frac{1}{2})}^{2} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{2 \frac{1^{2}}{2^{2}} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.3

Cancel the common factor of $2$ .

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Step 1.1.2.9.1.3.1

Factor $2$ out of $2^{2}$ .

$\frac{2 \frac{1^{2}}{2 \cdot 2} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.3.2

Cancel the common factor.

$\frac{2 \frac{1^{2}}{2 \cdot 2} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.3.3

Rewrite the expression.

$\frac{\frac{1^{2}}{2} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.4

One to any power is one.

$\frac{\frac{1}{2} + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.5

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

$\frac{\frac{1}{2} + 3 (\frac{1}{2}) - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.6

Combine $3$ and $\frac{1}{2}$ .

$\frac{\frac{1}{2} + \frac{3}{2} - 1 \cdot 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.1.7

Multiply $- 1$ by $2$ .

$\frac{\frac{1}{2} + \frac{3}{2} - 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.2

Combine the numerators over the common denominator.

$\frac{- 2 + \frac{1 + 3}{2}}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.3

Add $1$ and $3$ .

$\frac{- 2 + \frac{4}{2}}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.4

Divide $4$ by $2$ .

$\frac{- 2 + 2}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.2.9.5

Add $- 2$ and $2$ .

$\frac{0}{lim_{x \to \frac{π}{3}} 2 \cos (x) - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to \frac{π}{3}} 2 \cos (x) - lim_{x \to \frac{π}{3}} 1}$

Step 1.1.3.1.2

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

$\frac{0}{2 lim_{x \to \frac{π}{3}} \cos (x) - lim_{x \to \frac{π}{3}} 1}$

Step 1.1.3.1.3

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

$\frac{0}{2 \cos (lim_{x \to \frac{π}{3}} x) - lim_{x \to \frac{π}{3}} 1}$

Step 1.1.3.1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{3}$ .

$\frac{0}{2 \cos (lim_{x \to \frac{π}{3}} x) - 1 \cdot 1}$

Step 1.1.3.2

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

$\frac{0}{2 \cos (\frac{π}{3}) - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

$\frac{0}{2 (\frac{1}{2}) - 1 \cdot 1}$

Step 1.1.3.3.1.2

Cancel the common factor of $2$ .

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Step 1.1.3.3.1.2.1

Cancel the common factor.

$\frac{0}{2 (\frac{1}{2}) - 1 \cdot 1}$

Step 1.1.3.3.1.2.2

Rewrite the expression.

$\frac{0}{1 - 1 \cdot 1}$

Step 1.1.3.3.1.3

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{π}{3}} \frac{2 \cos^{2} (x) + 3 \cos (x) - 2}{2 \cos (x) - 1} = lim_{x \to \frac{π}{3}} \frac{\frac{d}{d x} [2 \cos^{2} (x) + 3 \cos (x) - 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{3}} \frac{\frac{d}{d x} [2 \cos^{2} (x) + 3 \cos (x) - 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 \cos^{2} (x) + 3 \cos (x) - 2$ with respect to $x$ is $\frac{d}{d x} [2 \cos^{2} (x)] + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]$ .

$lim_{x \to \frac{π}{3}} \frac{\frac{d}{d x} [2 \cos^{2} (x)] + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [2 \cos^{2} (x)]$ .

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Step 1.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \cos^{2} (x)$ with respect to $x$ is $2 \frac{d}{d x} [\cos^{2} (x)]$ .

$lim_{x \to \frac{π}{3}} \frac{2 \frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

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Step 1.3.3.2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$lim_{x \to \frac{π}{3}} \frac{2 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$lim_{x \to \frac{π}{3}} \frac{2 (2 u \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$lim_{x \to \frac{π}{3}} \frac{2 (2 \cos (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{3}} \frac{2 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.4

Multiply $- 1$ by $2$ .

$lim_{x \to \frac{π}{3}} \frac{2 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.3.5

Multiply $- 2$ by $2$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) + \frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 1.3.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) + 3 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.4.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) + 3 (- \sin (x)) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.4.3

Multiply $- 1$ by $3$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x) + 0}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.6

Add $- 4 \cos (x) \sin (x) - 3 \sin (x)$ and $0$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{\frac{d}{d x} [2 \cos (x) - 1]}$

Step 1.3.7

By the Sum Rule, the derivative of $2 \cos (x) - 1$ with respect to $x$ is $\frac{d}{d x} [2 \cos (x)] + \frac{d}{d x} [- 1]$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{\frac{d}{d x} [2 \cos (x)] + \frac{d}{d x} [- 1]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [2 \cos (x)]$ .

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Step 1.3.8.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \cos (x)$ with respect to $x$ is $2 \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{2 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 1]}$

Step 1.3.8.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{2 (- \sin (x)) + \frac{d}{d x} [- 1]}$

Step 1.3.8.3

Multiply $- 1$ by $2$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{- 2 \sin (x) + \frac{d}{d x} [- 1]}$

Step 1.3.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{- 2 \sin (x) + 0}$

Step 1.3.10

Add $- 2 \sin (x)$ and $0$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{- 2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- 2} lim_{x \to \frac{π}{3}} \frac{- 4 \cos (x) \sin (x) - 3 \sin (x)}{\sin (x)}$

Step 2.2

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

$\frac{1}{- 2} \cdot \frac{lim_{x \to \frac{π}{3}} - 4 \cos (x) \sin (x) - 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.3

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

$\frac{1}{- 2} \cdot \frac{- lim_{x \to \frac{π}{3}} 4 \cos (x) \sin (x) - lim_{x \to \frac{π}{3}} 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.4

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

$\frac{1}{- 2} \cdot \frac{- 4 lim_{x \to \frac{π}{3}} \cos (x) \sin (x) - lim_{x \to \frac{π}{3}} 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.5

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

$\frac{1}{- 2} \cdot \frac{- 4 lim_{x \to \frac{π}{3}} \cos (x) \cdot lim_{x \to \frac{π}{3}} \sin (x) - lim_{x \to \frac{π}{3}} 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.6

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (lim_{x \to \frac{π}{3}} x) \cdot lim_{x \to \frac{π}{3}} \sin (x) - lim_{x \to \frac{π}{3}} 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.7

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (lim_{x \to \frac{π}{3}} x) \cdot \sin (lim_{x \to \frac{π}{3}} x) - lim_{x \to \frac{π}{3}} 3 \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.8

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (lim_{x \to \frac{π}{3}} x) \cdot \sin (lim_{x \to \frac{π}{3}} x) - 3 lim_{x \to \frac{π}{3}} \sin (x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.9

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (lim_{x \to \frac{π}{3}} x) \cdot \sin (lim_{x \to \frac{π}{3}} x) - 3 \sin (lim_{x \to \frac{π}{3}} x)}{lim_{x \to \frac{π}{3}} \sin (x)}$

Step 2.10

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (lim_{x \to \frac{π}{3}} x) \cdot \sin (lim_{x \to \frac{π}{3}} x) - 3 \sin (lim_{x \to \frac{π}{3}} x)}{\sin (lim_{x \to \frac{π}{3}} x)}$

Step 3

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

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Step 3.1

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (\frac{π}{3}) \cdot \sin (lim_{x \to \frac{π}{3}} x) - 3 \sin (lim_{x \to \frac{π}{3}} x)}{\sin (lim_{x \to \frac{π}{3}} x)}$

Step 3.2

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (\frac{π}{3}) \cdot \sin (\frac{π}{3}) - 3 \sin (lim_{x \to \frac{π}{3}} x)}{\sin (lim_{x \to \frac{π}{3}} x)}$

Step 3.3

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

$\frac{1}{- 2} \cdot \frac{- 4 \cos (\frac{π}{3}) \cdot \sin (\frac{π}{3}) - 3 \sin (\frac{π}{3})}{\sin (lim_{x \to \frac{π}{3}} x)}$

Step 3.4

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

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

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Factor $2$ out of $- 4$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Factor $2$ out of $- 2$ .

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

Step 4.2.4.2

Cancel the common factor.

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

Step 4.2.4.3

Rewrite the expression.

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

Step 4.2.5

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 4.2.6

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

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

Step 4.2.7

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

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

Step 4.2.8

Move the negative in front of the fraction.

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

Step 4.2.9

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

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

Step 4.2.10

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

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

Step 4.2.11

Combine the numerators over the common denominator.

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

Step 4.2.12

Rewrite $\frac{- \sqrt{3} \cdot 2 - 3 \sqrt{3}}{2}$ in a factored form.

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Step 4.2.12.1

Multiply $2$ by $- 1$ .

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

Step 4.2.12.2

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

$- \frac{1}{2} \cdot \frac{\frac{- 5 \sqrt{3}}{2}}{\sin (\frac{π}{3})}$

Step 4.2.13

Move the negative in front of the fraction.

$- \frac{1}{2} \cdot \frac{- \frac{5 \sqrt{3}}{2}}{\sin (\frac{π}{3})}$

Step 4.3

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

$- \frac{1}{2} \cdot \frac{- \frac{5 \sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{2} (- \frac{5 \sqrt{3}}{2} \cdot \frac{2}{\sqrt{3}})$

Step 4.5

Cancel the common factor of $\sqrt{3}$ .

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Step 4.5.1

Move the leading negative in $- \frac{5 \sqrt{3}}{2}$ into the numerator.

$- \frac{1}{2} (\frac{- 5 \sqrt{3}}{2} \cdot \frac{2}{\sqrt{3}})$

Step 4.5.2

Factor $\sqrt{3}$ out of $- 5 \sqrt{3}$ .

$- \frac{1}{2} (\frac{\sqrt{3} \cdot - 5}{2} \cdot \frac{2}{\sqrt{3}})$

Step 4.5.3

Cancel the common factor.

$- \frac{1}{2} (\frac{\sqrt{3} \cdot - 5}{2} \cdot \frac{2}{\sqrt{3}})$

Step 4.5.4

Rewrite the expression.

$- \frac{1}{2} (\frac{- 5}{2} \cdot 2)$

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Cancel the common factor.

$- \frac{1}{2} (\frac{- 5}{2} \cdot 2)$

Step 4.6.2

Rewrite the expression.

$- \frac{1}{2} \cdot - 5$

Step 4.7

Multiply $- \frac{1}{2} \cdot - 5$ .

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Step 4.7.1

Multiply $- 5$ by $- 1$ .

$5 (\frac{1}{2})$

Step 4.7.2

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

$\frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$