Calculus Examples

$lim_{x \to 9} \frac{3 - \sqrt{x}}{27 - \sqrt{x^{3}}}$

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.

$\frac{lim_{x \to 9} 3 - \sqrt{x}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

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 $9$ .

$\frac{lim_{x \to 9} 3 - lim_{x \to 9} \sqrt{x}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.1.2

Evaluate the limit of $3$ which is constant as $x$ approaches $9$ .

$\frac{3 - lim_{x \to 9} \sqrt{x}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.1.3

Move the limit under the radical sign.

$\frac{3 - \sqrt{lim_{x \to 9} x}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{3 - \sqrt{9}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

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

Rewrite $9$ as $3^{2}$ .

$\frac{3 - \sqrt{3^{2}}}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.3.1.2

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

$\frac{3 - 1 \cdot 3}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.3.1.3

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

Step 1.1.2.3.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to 9} 27 - \sqrt{x^{3}}}$

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 $9$ .

$\frac{0}{lim_{x \to 9} 27 - lim_{x \to 9} \sqrt{x^{3}}}$

Step 1.1.3.1.2

Evaluate the limit of $27$ which is constant as $x$ approaches $9$ .

$\frac{0}{27 - lim_{x \to 9} \sqrt{x^{3}}}$

Step 1.1.3.1.3

Move the limit under the radical sign.

$\frac{0}{27 - \sqrt{lim_{x \to 9} x^{3}}}$

Step 1.1.3.1.4

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

$\frac{0}{27 - \sqrt{{(lim_{x \to 9} x)}^{3}}}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{0}{27 - \sqrt{9^{3}}}$

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

Raise $9$ to the power of $3$ .

$\frac{0}{27 - \sqrt{729}}$

Step 1.1.3.3.1.2

Rewrite $729$ as $27^{2}$ .

$\frac{0}{27 - \sqrt{27^{2}}}$

Step 1.1.3.3.1.3

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

$\frac{0}{27 - 1 \cdot 27}$

Step 1.1.3.3.1.4

Multiply $- 1$ by $27$ .

$\frac{0}{27 - 27}$

Step 1.1.3.3.2

Subtract $27$ from $27$ .

$\frac{0}{0}$

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.

$lim_{x \to 9} \frac{3 - \sqrt{x}}{27 - \sqrt{x^{3}}} = lim_{x \to 9} \frac{\frac{d}{d x} [3 - \sqrt{x}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 9} \frac{\frac{d}{d x} [3 - \sqrt{x}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.2

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

$lim_{x \to 9} \frac{\frac{d}{d x} [3] + \frac{d}{d x} [- \sqrt{x}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{0 + \frac{d}{d x} [- \sqrt{x}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 9} \frac{0 + \frac{d}{d x} [- x^{\frac{1}{2}}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.2

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

$lim_{x \to 9} \frac{0 - \frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{1}{2} - 1})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.5

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

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.6

Combine the numerators over the common denominator.

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.7

Simplify the numerator.

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Step 1.3.4.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{1 - 2}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.7.2

Subtract $2$ from $1$ .

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{\frac{- 1}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.8

Move the negative in front of the fraction.

$lim_{x \to 9} \frac{0 - (\frac{1}{2} x^{- \frac{1}{2}})}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.9

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

$lim_{x \to 9} \frac{0 - \frac{x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.4.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 9} \frac{0 - \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.5

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from $0$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [27 - \sqrt{x^{3}}]}$

Step 1.3.6

By the Sum Rule, the derivative of $27 - \sqrt{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [27] + \frac{d}{d x} [- \sqrt{x^{3}}]$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [27] + \frac{d}{d x} [- \sqrt{x^{3}}]}$

Step 1.3.7

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 + \frac{d}{d x} [- \sqrt{x^{3}}]}$

Step 1.3.8

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

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Step 1.3.8.1

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

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 + \frac{d}{d x} [- x^{\frac{3}{2}}]}$

Step 1.3.8.2

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

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - \frac{d}{d x} [x^{\frac{3}{2}}]}$

Step 1.3.8.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{3}{2} - 1})}$

Step 1.3.8.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})}$

Step 1.3.8.5

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

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})}$

Step 1.3.8.6

Combine the numerators over the common denominator.

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})}$

Step 1.3.8.7

Simplify the numerator.

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Step 1.3.8.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{3 - 2}{2}})}$

Step 1.3.8.7.2

Subtract $2$ from $3$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - (\frac{3}{2} x^{\frac{1}{2}})}$

Step 1.3.8.8

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

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{0 - \frac{3 x^{\frac{1}{2}}}{2}}$

Step 1.3.9

Subtract $\frac{3 x^{\frac{1}{2}}}{2}$ from $0$ .

$lim_{x \to 9} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{- \frac{3 x^{\frac{1}{2}}}{2}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 9} - \frac{1}{2 x^{\frac{1}{2}}} (- \frac{2}{3 x^{\frac{1}{2}}})$

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 9} - \frac{1}{2 \sqrt{x}} (- \frac{2}{3 x^{\frac{1}{2}}})$

Step 1.5.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 9} - \frac{1}{2 \sqrt{x}} (- \frac{2}{3 \sqrt{x}})$

Step 1.6

Combine factors.

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Step 1.6.1

Multiply $- 1$ by $- 1$ .

$lim_{x \to 9} 1 \frac{1}{2 \sqrt{x}} \frac{2}{3 \sqrt{x}}$

Step 1.6.2

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

$lim_{x \to 9} \frac{1}{2 \sqrt{x}} \cdot \frac{2}{3 \sqrt{x}}$

Step 1.6.3

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

$lim_{x \to 9} \frac{2}{2 \sqrt{x} (3 \sqrt{x})}$

Step 1.6.4

Multiply $3$ by $2$ .

$lim_{x \to 9} \frac{2}{6 \sqrt{x} \sqrt{x}}$

Step 1.6.5

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

$lim_{x \to 9} \frac{2}{6 ({\sqrt{x}}^{1} \sqrt{x})}$

Step 1.6.6

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

$lim_{x \to 9} \frac{2}{6 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 1.6.7

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

$lim_{x \to 9} \frac{2}{6 {\sqrt{x}}^{1 + 1}}$

Step 1.6.8

Add $1$ and $1$ .

$lim_{x \to 9} \frac{2}{6 {\sqrt{x}}^{2}}$

Step 1.7

Cancel the common factor of $2$ and $6$ .

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Step 1.7.1

Factor $2$ out of $2$ .

$lim_{x \to 9} \frac{2 (1)}{6 {\sqrt{x}}^{2}}$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $2$ out of $6 {\sqrt{x}}^{2}$ .

$lim_{x \to 9} \frac{2 (1)}{2 (3 {\sqrt{x}}^{2})}$

Step 1.7.2.2

Cancel the common factor.

$lim_{x \to 9} \frac{2 \cdot 1}{2 (3 {\sqrt{x}}^{2})}$

Step 1.7.2.3

Rewrite the expression.

$lim_{x \to 9} \frac{1}{3 {\sqrt{x}}^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 9} \frac{1}{{\sqrt{x}}^{2}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $9$ .

$\frac{1}{3} \cdot \frac{lim_{x \to 9} 1}{lim_{x \to 9} {\sqrt{x}}^{2}}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $9$ .

$\frac{1}{3} \cdot \frac{1}{lim_{x \to 9} {\sqrt{x}}^{2}}$

Step 2.4

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

$\frac{1}{3} \cdot \frac{1}{{(lim_{x \to 9} \sqrt{x})}^{2}}$

Step 2.5

Move the limit under the radical sign.

$\frac{1}{3} \cdot \frac{1}{{\sqrt{lim_{x \to 9} x}}^{2}}$

Step 3

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{1}{3} \cdot \frac{1}{{\sqrt{9}}^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{1 \cdot 1}{3 {\sqrt{9}}^{2}}$

Step 4.2

Multiply $1$ by $1$ .

$\frac{1}{3 {\sqrt{9}}^{2}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.3.2

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

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

Step 4.3.3

Raise $3$ to the power of $2$ .

$\frac{1}{3 \cdot 9}$

Step 4.4

Multiply $3$ by $9$ .

$\frac{1}{27}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{27}$

Decimal Form:

$0.\overline{037}$