Calculus Examples

$lim_{x \to \infty} x^{3} \sin (\frac{1}{2 x^{3}})$

Step 1

Rewrite $x^{3} \sin (\frac{1}{2 x^{3}})$ as $\frac{\sin (\frac{1}{2 x^{3}})}{x^{- 3}}$ .

$lim_{x \to \infty} \frac{\sin (\frac{1}{2 x^{3}})}{x^{- 3}}$

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.

$\frac{lim_{x \to \infty} \sin (\frac{1}{2 x^{3}})}{lim_{x \to \infty} x^{- 3}}$

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.

$\frac{\sin (lim_{x \to \infty} \frac{1}{2 x^{3}})}{lim_{x \to \infty} x^{- 3}}$

Step 2.1.2.1.2

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

$\frac{\sin (\frac{1}{2} lim_{x \to \infty} \frac{1}{x^{3}})}{lim_{x \to \infty} x^{- 3}}$

Step 2.1.2.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $\frac{1}{2}$ by $0$ .

$\frac{\sin (0)}{lim_{x \to \infty} x^{- 3}}$

Step 2.1.2.3.2

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

$\frac{0}{lim_{x \to \infty} x^{- 3}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$\frac{0}{lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 2.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{0}{0}$

Step 2.1.3.3

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

Undefined

$\frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

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 $g (x) = \frac{1}{2 x^{3}}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $\frac{1}{2 x^{3}}$ .

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Replace all occurrences of $u$ with $\frac{1}{2 x^{3}}$ .

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2 x^{3}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 2.3.4

Combine $\frac{1}{2}$ and $\cos (\frac{1}{2 x^{3}})$ .

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

Step 2.3.5

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 2.3.6

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.3.6.1

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

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

Step 2.3.6.2

Multiply $3$ by $- 1$ .

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

Step 2.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

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

Step 2.3.8

Combine $- 3$ and $\frac{\cos (\frac{1}{2 x^{3}})}{2}$ .

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

Step 2.3.9

Combine $\frac{- 3 \cos (\frac{1}{2 x^{3}})}{2}$ and $x^{- 4}$ .

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

Step 2.3.10

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.11

Move the negative in front of the fraction.

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

Step 2.3.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

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

Step 2.3.13

Simplify.

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Step 2.3.13.1

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

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

Step 2.3.13.2

Combine $- 3$ and $\frac{1}{x^{4}}$ .

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

Step 2.3.13.3

Move the negative in front of the fraction.

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Combine factors.

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

$lim_{x \to \infty} 1 \frac{3 \cos (\frac{1}{2 x^{3}})}{2 x^{4}} \frac{x^{4}}{3}$

Step 2.5.2

Multiply $\frac{3 \cos (\frac{1}{2 x^{3}})}{2 x^{4}}$ by $1$ .

$lim_{x \to \infty} \frac{3 \cos (\frac{1}{2 x^{3}})}{2 x^{4}} \cdot \frac{x^{4}}{3}$

Step 2.5.3

Multiply $\frac{3 \cos (\frac{1}{2 x^{3}})}{2 x^{4}}$ by $\frac{x^{4}}{3}$ .

$lim_{x \to \infty} \frac{3 \cos (\frac{1}{2 x^{3}}) x^{4}}{2 x^{4} \cdot 3}$

Step 2.5.4

Multiply $3$ by $2$ .

$lim_{x \to \infty} \frac{3 \cos (\frac{1}{2 x^{3}}) x^{4}}{6 x^{4}}$

Step 2.6

Reduce.

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Step 2.6.1

Cancel the common factor of $3$ and $6$ .

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Step 2.6.1.1

Factor $3$ out of $3 \cos (\frac{1}{2 x^{3}}) x^{4}$ .

$lim_{x \to \infty} \frac{3 (\cos (\frac{1}{2 x^{3}}) x^{4})}{6 x^{4}}$

Step 2.6.1.2

Cancel the common factors.

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Step 2.6.1.2.1

Factor $3$ out of $6 x^{4}$ .

$lim_{x \to \infty} \frac{3 (\cos (\frac{1}{2 x^{3}}) x^{4})}{3 (2 x^{4})}$

Step 2.6.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{3 (\cos (\frac{1}{2 x^{3}}) x^{4})}{3 (2 x^{4})}$

Step 2.6.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\cos (\frac{1}{2 x^{3}}) x^{4}}{2 x^{4}}$

Step 2.6.2

Cancel the common factor of $x^{4}$ .

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Step 2.6.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\cos (\frac{1}{2 x^{3}}) x^{4}}{2 x^{4}}$

Step 2.6.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\cos (\frac{1}{2 x^{3}})}{2}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{1}{2} lim_{x \to \infty} \cos (\frac{1}{2 x^{3}})$

Step 3.2

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

$\frac{1}{2} \cos (lim_{x \to \infty} \frac{1}{2 x^{3}})$

Step 3.3

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

$\frac{1}{2} \cos (\frac{1}{2} lim_{x \to \infty} \frac{1}{x^{3}})$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

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

Step 5

Simplify the answer.

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Step 5.1

Multiply $\frac{1}{2}$ by $0$ .

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

Step 5.2

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

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

Step 5.3

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$