Calculus Examples

$lim_{x \to 0} \frac{\sin (x)}{\sqrt[3]{x}}$

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.

$\frac{lim_{x \to 0} \sin (x)}{lim_{x \to 0} \sqrt[3]{x}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{\sin (lim_{x \to 0} x)}{lim_{x \to 0} \sqrt[3]{x}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (0)}{lim_{x \to 0} \sqrt[3]{x}}$

Step 1.1.2.3

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

$\frac{0}{lim_{x \to 0} \sqrt[3]{x}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Move the limit under the radical sign.

$\frac{0}{\sqrt[3]{lim_{x \to 0} x}}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\sqrt[3]{0}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Rewrite $0$ as $0^{3}$ .

$\frac{0}{\sqrt[3]{0^{3}}}$

Step 1.1.3.3.2

Pull terms out from under the radical, assuming real numbers.

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sin (x)}{\sqrt[3]{x}} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [\sqrt[3]{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

$lim_{x \to 0} \frac{\cos (x)}{\frac{d}{d x} [x^{\frac{1}{3}}]}$

Step 1.3.4

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

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

Step 1.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.3.6

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

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

Step 1.3.7

Combine the numerators over the common denominator.

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

Step 1.3.8

Simplify the numerator.

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Step 1.3.8.1

Multiply $- 1$ by $3$ .

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

Step 1.3.8.2

Subtract $3$ from $1$ .

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

Step 1.3.9

Move the negative in front of the fraction.

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

Step 1.3.10

Simplify.

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Step 1.3.10.1

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

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

Step 1.3.10.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \cos (x) (3 x^{\frac{2}{3}})$

Step 2

Evaluate the limit.

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Step 2.1

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

$3 lim_{x \to 0} \cos (x) x^{\frac{2}{3}}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$3 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} x^{\frac{2}{3}}$

Step 2.3

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

$3 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} x^{\frac{2}{3}}$

Step 2.4

Move the exponent $\frac{2}{3}$ from $x^{\frac{2}{3}}$ outside the limit using the Limits Power Rule.

$3 \cos (lim_{x \to 0} x) \cdot {(lim_{x \to 0} x)}^{\frac{2}{3}}$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$3 \cos (0) \cdot {(lim_{x \to 0} x)}^{\frac{2}{3}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$3 \cos (0) \cdot 0^{\frac{2}{3}}$

Step 4

Simplify the answer.

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Step 4.1

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

$3 \cdot 1 \cdot 0^{\frac{2}{3}}$

Step 4.2

Multiply $3$ by $1$ .

$3 \cdot 0^{\frac{2}{3}}$

Step 4.3

Rewrite $0$ as $0^{3}$ .

$3 \cdot {(0^{3})}^{\frac{2}{3}}$

Step 4.4

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

$3 \cdot 0^{3 (\frac{2}{3})}$

Step 4.5

Cancel the common factor of $3$ .

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Step 4.5.1

Cancel the common factor.

$3 \cdot 0^{3 (\frac{2}{3})}$

Step 4.5.2

Rewrite the expression.

$3 \cdot 0^{2}$

Step 4.6

Raising $0$ to any positive power yields $0$ .

$3 \cdot 0$

Step 4.7

Multiply $3$ by $0$ .

$0$