Calculus Examples

$lim_{x \to 81} \frac{x (\sqrt{x} - 9)}{x - 81}$

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.

$\frac{lim_{x \to 81} x (\sqrt{x} - 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x \cdot lim_{x \to 81} \sqrt{x} - 9}{lim_{x \to 81} x - 81}$

Step 1.1.2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x \cdot (lim_{x \to 81} \sqrt{x} - lim_{x \to 81} 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.3

Move the limit under the radical sign.

$\frac{lim_{x \to 81} x \cdot (\sqrt{lim_{x \to 81} x} - lim_{x \to 81} 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.4

Evaluate the limit of $9$ which is constant as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x \cdot (\sqrt{lim_{x \to 81} x} - 1 \cdot 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.5

Evaluate the limits by plugging in $81$ for all occurrences of $x$ .

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Step 1.1.2.5.1

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

$\frac{81 \cdot (\sqrt{lim_{x \to 81} x} - 1 \cdot 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.5.2

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

$\frac{81 \cdot (\sqrt{81} - 1 \cdot 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Rewrite $81$ as $9^{2}$ .

$\frac{81 \cdot (\sqrt{9^{2}} - 1 \cdot 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.6.1.2

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

$\frac{81 \cdot (9 - 1 \cdot 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.6.1.3

Multiply $- 1$ by $9$ .

$\frac{81 \cdot (9 - 9)}{lim_{x \to 81} x - 81}$

Step 1.1.2.6.2

Subtract $9$ from $9$ .

$\frac{81 \cdot 0}{lim_{x \to 81} x - 81}$

Step 1.1.2.6.3

Multiply $81$ by $0$ .

$\frac{0}{lim_{x \to 81} x - 81}$

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 $81$ .

$\frac{0}{lim_{x \to 81} x - lim_{x \to 81} 81}$

Step 1.1.3.1.2

Evaluate the limit of $81$ which is constant as $x$ approaches $81$ .

$\frac{0}{lim_{x \to 81} x - 1 \cdot 81}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $81$ .

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

Step 1.1.3.3.2

Subtract $81$ from $81$ .

$\frac{0}{0}$

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.

$lim_{x \to 81} \frac{x (\sqrt{x} - 9)}{x - 81} = lim_{x \to 81} \frac{\frac{d}{d x} [x (\sqrt{x} - 9)]}{\frac{d}{d x} [x - 81]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 81} \frac{\frac{d}{d x} [x (\sqrt{x} - 9)]}{\frac{d}{d x} [x - 81]}$

Step 1.3.2

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

$lim_{x \to 81} \frac{\frac{d}{d x} [x (x^{\frac{1}{2}} - 9)]}{\frac{d}{d x} [x - 81]}$

Step 1.3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{\frac{1}{2}} - 9$ .

$lim_{x \to 81} \frac{x \frac{d}{d x} [x^{\frac{1}{2}} - 9] + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.4

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 9]$ .

$lim_{x \to 81} \frac{x (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.5

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

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.6

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

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.7

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

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.8

Combine the numerators over the common denominator.

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.9

Simplify the numerator.

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Step 1.3.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.9.2

Subtract $2$ from $1$ .

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.10

Move the negative in front of the fraction.

$lim_{x \to 81} \frac{x (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.11

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

$lim_{x \to 81} \frac{x (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.12

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

$lim_{x \to 81} \frac{x (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 9]) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.13

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$lim_{x \to 81} \frac{x (\frac{1}{2 x^{\frac{1}{2}}} + 0) + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.14

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 81} \frac{x \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.15

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

$lim_{x \to 81} \frac{\frac{x}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.16

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

$lim_{x \to 81} \frac{\frac{x \cdot x^{- \frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.17

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.3.17.1

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

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Step 1.3.17.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 81} \frac{\frac{x^{1} x^{- \frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.17.1.2

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

$lim_{x \to 81} \frac{\frac{x^{1 - \frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.17.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 81} \frac{\frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.17.3

Combine the numerators over the common denominator.

$lim_{x \to 81} \frac{\frac{x^{\frac{2 - 1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.17.4

Subtract $1$ from $2$ .

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \frac{d}{d x} [x]}{\frac{d}{d x} [x - 81]}$

Step 1.3.18

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

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}}}{2} + (x^{\frac{1}{2}} - 9) \cdot 1}{\frac{d}{d x} [x - 81]}$

Step 1.3.19

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

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}}}{2} + x^{\frac{1}{2}} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.20

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

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}}}{2} + x^{\frac{1}{2}} \cdot \frac{2}{2} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.21

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

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}}}{2} + \frac{x^{\frac{1}{2}} \cdot 2}{2} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.22

Combine the numerators over the common denominator.

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot 2}{2} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.23

Move $2$ to the left of $x^{\frac{1}{2}}$ .

$lim_{x \to 81} \frac{\frac{x^{\frac{1}{2}} + 2 \cdot x^{\frac{1}{2}}}{2} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.24

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

$lim_{x \to 81} \frac{\frac{3 x^{\frac{1}{2}}}{2} - 9}{\frac{d}{d x} [x - 81]}$

Step 1.3.25

By the Sum Rule, the derivative of $x - 81$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 81]$ .

$lim_{x \to 81} \frac{\frac{3 x^{\frac{1}{2}}}{2} - 9}{\frac{d}{d x} [x] + \frac{d}{d x} [- 81]}$

Step 1.3.26

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

$lim_{x \to 81} \frac{\frac{3 x^{\frac{1}{2}}}{2} - 9}{1 + \frac{d}{d x} [- 81]}$

Step 1.3.27

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$lim_{x \to 81} \frac{\frac{3 x^{\frac{1}{2}}}{2} - 9}{1 + 0}$

Step 1.3.28

Add $1$ and $0$ .

$lim_{x \to 81} \frac{\frac{3 x^{\frac{1}{2}}}{2} - 9}{1}$

Step 1.4

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

$lim_{x \to 81} \frac{\frac{3 \sqrt{x}}{2} - 9}{1}$

Step 1.5

Combine terms.

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Step 1.5.1

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 81} \frac{\frac{3 \sqrt{x}}{2} - 9 \cdot \frac{2}{2}}{1}$

Step 1.5.2

Combine $- 9$ and $\frac{2}{2}$ .

$lim_{x \to 81} \frac{\frac{3 \sqrt{x}}{2} + \frac{- 9 \cdot 2}{2}}{1}$

Step 1.5.3

Combine the numerators over the common denominator.

$lim_{x \to 81} \frac{\frac{3 \sqrt{x} - 9 \cdot 2}{2}}{1}$

Step 1.6

Divide $\frac{3 \sqrt{x} - 9 \cdot 2}{2}$ by $1$ .

$lim_{x \to 81} \frac{3 \sqrt{x} - 9 \cdot 2}{2}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{1}{2} lim_{x \to 81} 3 \sqrt{x} - 9 \cdot 2$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $81$ .

$\frac{1}{2} (lim_{x \to 81} 3 \sqrt{x} - lim_{x \to 81} 9 \cdot 2)$

Step 2.3

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

$\frac{1}{2} (3 lim_{x \to 81} \sqrt{x} - lim_{x \to 81} 9 \cdot 2)$

Step 2.4

Move the limit under the radical sign.

$\frac{1}{2} (3 \sqrt{lim_{x \to 81} x} - lim_{x \to 81} 9 \cdot 2)$

Step 2.5

Evaluate the limit of $9 \cdot 2$ which is constant as $x$ approaches $81$ .

$\frac{1}{2} (3 \sqrt{lim_{x \to 81} x} - 9 \cdot 2)$

Step 3

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

$\frac{1}{2} (3 \sqrt{81} - 9 \cdot 2)$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite $81$ as $9^{2}$ .

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

Step 4.1.2

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

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

Step 4.1.3

Multiply $3$ by $9$ .

$\frac{1}{2} (27 - 9 \cdot 2)$

Step 4.1.4

Multiply $- 9$ by $2$ .

$\frac{1}{2} (27 - 18)$

Step 4.2

Subtract $18$ from $27$ .

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

Step 4.3

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$