Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.1.4

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $- 2$ .

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

Step 1.1.2.3.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.2.3

Rewrite the expression.

$\frac{1 - 1}{lim_{x \to \frac{π}{3}} π - 3 x}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to \frac{π}{3}} π - 3 x}$

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 $\frac{π}{3}$ .

$\frac{0}{lim_{x \to \frac{π}{3}} π - lim_{x \to \frac{π}{3}} 3 x}$

Step 1.1.3.1.2

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

$\frac{0}{π - lim_{x \to \frac{π}{3}} 3 x}$

Step 1.1.3.1.3

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

$\frac{0}{π - 3 lim_{x \to \frac{π}{3}} x}$

Step 1.1.3.2

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

$\frac{0}{π - 3 \frac{π}{3}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Cancel the common factor of $3$ .

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Step 1.1.3.3.1.1

Factor $3$ out of $- 3$ .

$\frac{0}{π + 3 (- 1) \frac{π}{3}}$

Step 1.1.3.3.1.2

Cancel the common factor.

$\frac{0}{π + 3 \cdot - 1 \frac{π}{3}}$

Step 1.1.3.3.1.3

Rewrite the expression.

$\frac{0}{π - π}$

Step 1.1.3.3.2

Subtract $π$ from $π$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $- 2$ .

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

Multiply $- 3$ by $1$ .

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

Step 1.3.9

Subtract $3$ from $0$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{2}{- 3}$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.2

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

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

Step 3

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

$\frac{2}{- 3} \sin (\frac{π}{3})$

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{2}{3} \sin (\frac{π}{3})$

Step 4.2

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

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

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

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

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

Step 4.3.2

Factor $2$ out of $- 2$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

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

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.57735026 \dots$