Calculus Examples

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

Step 1

Rewrite $x \sin (\frac{2 π}{x})$ as $\frac{\sin (\frac{2 π}{x})}{\frac{1}{x}}$ .

$lim_{x \to \infty} \frac{\sin (\frac{2 π}{x})}{\frac{1}{x}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to \infty} \sin (\frac{2 π}{x})}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$\frac{\sin (lim_{x \to \infty} \frac{2 π}{x})}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.2.1.2

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

$\frac{\sin (2 π lim_{x \to \infty} \frac{1}{x})}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.2.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $2 π \cdot 0$ .

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Step 2.1.2.3.1.1

Multiply $0$ by $2$ .

$\frac{\sin (0 π)}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.2.3.1.2

Multiply $0$ by $π$ .

$\frac{\sin (0)}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.2.3.2

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

$\frac{0}{lim_{x \to \infty} \frac{1}{x}}$

Step 2.1.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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 $g (x) = \frac{2 π}{x}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $\frac{2 π}{x}$ .

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Replace all occurrences of $u$ with $\frac{2 π}{x}$ .

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

Step 2.3.3

Since $2 π$ is constant with respect to $x$ , the derivative of $\frac{2 π}{x}$ with respect to $x$ is $2 π \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.3.4

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $- 1$ by $2$ .

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.7.2

Combine terms.

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Step 2.3.7.2.1

Combine $\frac{1}{x^{2}}$ and $- 2$ .

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

Step 2.3.7.2.2

Combine $\frac{- 2}{x^{2}}$ and $π$ .

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

Step 2.3.7.2.3

Move the negative in front of the fraction.

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

Step 2.3.7.2.4

Combine $\cos (\frac{2 π}{x})$ and $\frac{2 π}{x^{2}}$ .

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

Step 2.3.7.2.5

Move $2$ to the left of $\cos (\frac{2 π}{x})$ .

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

Step 2.3.7.3

Reorder factors in $- \frac{2 \cos (\frac{2 π}{x}) π}{x^{2}}$ .

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

Step 2.3.8

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.3.9

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

$lim_{x \to \infty} \frac{- \frac{2 π \cos (\frac{2 π}{x})}{x^{2}}}{- x^{- 2}}$

Step 2.3.10

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to \infty} - \frac{2 π \cos (\frac{2 π}{x})}{x^{2}} (- x^{2})$

Step 2.5

Combine factors.

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

$lim_{x \to \infty} 1 \frac{2 π \cos (\frac{2 π}{x})}{x^{2}} x^{2}$

Step 2.5.2

Multiply $\frac{2 π \cos (\frac{2 π}{x})}{x^{2}}$ by $1$ .

$lim_{x \to \infty} \frac{2 π \cos (\frac{2 π}{x})}{x^{2}} x^{2}$

Step 2.5.3

Combine $\frac{2 π \cos (\frac{2 π}{x})}{x^{2}}$ and $x^{2}$ .

$lim_{x \to \infty} \frac{2 π \cos (\frac{2 π}{x}) x^{2}}{x^{2}}$

Step 2.6

Cancel the common factor of $x^{2}$ .

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Step 2.6.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 π \cos (\frac{2 π}{x}) x^{2}}{x^{2}}$

Step 2.6.2

Divide $2 π \cos (\frac{2 π}{x})$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$2 π \cos (2 π \cdot 0)$

Step 5

Simplify the answer.

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Step 5.1

Multiply $2 π \cdot 0$ .

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Step 5.1.1

Multiply $0$ by $2$ .

$2 π \cos (0 π)$

Step 5.1.2

Multiply $0$ by $π$ .

$2 π \cos (0)$

Step 5.2

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

$2 π \cdot 1$

Step 5.3

Multiply $2$ by $1$ .

$2 π$