Calculus Examples

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

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.

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

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 $9$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Move the limit under the radical sign.

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

Step 1.1.2.2

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

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

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 $3^{2}$ .

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

Step 1.1.2.3.1.2

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

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

Step 1.1.2.3.1.3

Multiply $- 1$ by $3$ .

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

Step 1.1.2.3.2

Subtract $3$ from $3$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

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

$\frac{0}{9 \cdot 9 - 9^{2}}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

Multiply $9$ by $9$ .

$\frac{0}{81 - 9^{2}}$

Step 1.1.3.5.1.2

Raise $9$ to the power of $2$ .

$\frac{0}{81 - 1 \cdot 81}$

Step 1.1.3.5.1.3

Multiply $- 1$ by $81$ .

$\frac{0}{81 - 81}$

Step 1.1.3.5.2

Subtract $81$ from $81$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

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 $x^{\frac{1}{2}}$ .

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

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 $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

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 $n = \frac{1}{2}$ .

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

Step 1.3.4.4

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

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

Step 1.3.4.5

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

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

Step 1.3.4.6

Combine the numerators over the common denominator.

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

Step 1.3.4.7

Simplify the numerator.

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Step 1.3.4.7.1

Multiply $- 1$ by $2$ .

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

Step 1.3.4.7.2

Subtract $2$ from $1$ .

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

Step 1.3.4.8

Move the negative in front of the fraction.

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

Step 1.3.4.9

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

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

Step 1.3.4.10

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

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

Step 1.3.5

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

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

Step 1.3.6

By the Sum Rule, the derivative of $9 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.3.7

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.3.7.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

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

Step 1.3.7.2

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

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

Step 1.3.7.3

Multiply $9$ by $1$ .

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

Step 1.3.8

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

Multiply $2$ by $- 1$ .

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

Step 1.3.9

Reorder terms.

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

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

Step 1.6

Multiply $\frac{1}{- 2 x + 9}$ by $\frac{1}{2 \sqrt{x}}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Move the limit under the radical sign.

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

Step 3

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 3.1

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

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

Step 3.2

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

$- \frac{1}{2} \cdot \frac{1}{(- 2 \cdot 9 + 9) \cdot \sqrt{9}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $- 2$ by $9$ .

$- \frac{1}{2} \cdot \frac{1}{(- 18 + 9) \sqrt{9}}$

Step 4.1.2

Add $- 18$ and $9$ .

$- \frac{1}{2} \cdot \frac{1}{- 9 \sqrt{9}}$

Step 4.1.3

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

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

Step 4.1.4

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

$- \frac{1}{2} \cdot \frac{1}{- 9 \cdot 3}$

Step 4.2

Multiply $- 9$ by $3$ .

$- \frac{1}{2} \cdot \frac{1}{- 27}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{1}{2} (- \frac{1}{27})$

Step 4.4

Multiply $- \frac{1}{2} (- \frac{1}{27})$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{1}{27}$

Step 4.4.2

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

$\frac{1}{2} \cdot \frac{1}{27}$

Step 4.4.3

Multiply $\frac{1}{2}$ by $\frac{1}{27}$ .

$\frac{1}{2 \cdot 27}$

Step 4.4.4

Multiply $2$ by $27$ .

$\frac{1}{54}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{54}$

Decimal Form:

$0.0\overline{185}$