Calculus Examples

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

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.

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

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

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Cancel the common factor of $3$ .

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Step 1.1.2.3.1.1

Cancel the common factor.

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

Step 1.1.2.3.1.2

Rewrite the expression.

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

The exact value of $\sin (0)$ is $0$ .

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

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 $\frac{π}{3}$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.1.4

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

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

Step 1.1.3.2

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

$\frac{0}{1 - 2 \cos (\frac{π}{3})}$

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 $\frac{1}{2}$ .

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

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

Factor $2$ out of $- 2$ .

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

Step 1.1.3.3.1.2.2

Cancel the common factor.

$\frac{0}{1 + 2 \cdot - 1 \frac{1}{2}}$

Step 1.1.3.3.1.2.3

Rewrite the expression.

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Multiply $3$ by $1$ .

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

Step 1.3.6

Move $3$ to the left of $\cos (3 x)$ .

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

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

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Step 1.3.9.1

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

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

Step 1.3.9.2

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

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

Step 1.3.9.3

Multiply $- 1$ by $- 2$ .

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

Step 1.3.10

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

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

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 $x$ .

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{\cos (3 \frac{π}{3})}{\sin (\frac{π}{3})}$ as a product.

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

Step 4.2

Write $\cos (3 \frac{π}{3})$ as a fraction with denominator $1$ .

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

Step 4.3

Simplify.

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Step 4.3.1

Divide $\cos (3 \frac{π}{3})$ by $1$ .

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

Step 4.3.2

Convert from $\frac{1}{\sin (\frac{π}{3})}$ to $\csc (\frac{π}{3})$ .

$\frac{3}{2} (\cos (3 \frac{π}{3}) \csc (\frac{π}{3}))$

Step 4.4

Cancel the common factor of $3$ .

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Step 4.4.1

Cancel the common factor.

$\frac{3}{2} (\cos (3 \frac{π}{3}) \csc (\frac{π}{3}))$

Step 4.4.2

Rewrite the expression.

$\frac{3}{2} (\cos (π) \csc (\frac{π}{3}))$

Step 4.5

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

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

Step 4.6

The exact value of $\cos (0)$ is $1$ .

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

Step 4.7

Multiply $- 1$ by $1$ .

$\frac{3}{2} (- \csc (\frac{π}{3}))$

Step 4.8

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

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

Step 4.9

Cancel the common factor of $2$ .

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Step 4.9.1

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

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

Step 4.9.2

Factor $2$ out of $- 2$ .

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

Step 4.9.3

Cancel the common factor.

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

Step 4.9.4

Rewrite the expression.

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

Step 4.10

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

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

Step 4.11

Multiply $3$ by $- 1$ .

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

Step 4.12

Move the negative in front of the fraction.

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

Step 4.13

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

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

Step 4.14

Combine and simplify the denominator.

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Step 4.14.1

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

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

Step 4.14.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.14.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.14.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.14.5

Add $1$ and $1$ .

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

Step 4.14.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.14.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.14.6.3

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

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

Step 4.14.6.4

Cancel the common factor of $2$ .

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Step 4.14.6.4.1

Cancel the common factor.

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

Step 4.14.6.4.2

Rewrite the expression.

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

Step 4.14.6.5

Evaluate the exponent.

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

Step 4.15

Cancel the common factor of $3$ .

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Step 4.15.1

Cancel the common factor.

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

Step 4.15.2

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

$- \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$