Calculus Examples

lim x \to 81 \frac{\sqrt{x} - 9}{x - 81}

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

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 \sqrt{x} - 9}{lim x \to 81 x - 81}

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$x$ approaches

81

$81$ .

\frac{lim x \to 81 \sqrt{x} - lim x \to 81 9}{lim x \to 81 x - 81}

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

Step 1.1.2.1.2

Move the limit under the radical sign.

\frac{\sqrt{lim x \to 81 x} - lim x \to 81 9}{lim x \to 81 x - 81}

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

Step 1.1.2.1.3

Evaluate the limit of

9

$9$ which is constant as

x

$x$ approaches

81

$81$ .

\frac{\sqrt{lim x \to 81 x} - 1 \cdot 9}{lim x \to 81 x - 81}

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

\frac{\sqrt{lim x \to 81 x} - 1 \cdot 9}{lim x \to 81 x - 81}

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

Step 1.1.2.2

Evaluate the limit of

x

$x$ by plugging in

81

$81$ for

x

$x$ .

\frac{\sqrt{81} - 1 \cdot 9}{lim x \to 81 x - 81}

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

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

81

$81$ as

9^{2}

$9^{2}$ .

\frac{\sqrt{9^{2}} - 1 \cdot 9}{lim x \to 81 x - 81}

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

Step 1.1.2.3.1.2

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

\frac{9 - 1 \cdot 9}{lim x \to 81 x - 81}

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

Step 1.1.2.3.1.3

Multiply

- 1

$- 1$ by

9

$9$ .

\frac{9 - 9}{lim x \to 81 x - 81}

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

\frac{9 - 9}{lim x \to 81 x - 81}

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

Step 1.1.2.3.2

Subtract

9

$9$ from

9

$9$ .

\frac{0}{lim x \to 81 x - 81}

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

\frac{0}{lim x \to 81 x - 81}

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

\frac{0}{lim x \to 81 x - 81}

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

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

$x$ approaches

81

$81$ .

\frac{0}{lim x \to 81 x - lim x \to 81 81}

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

Step 1.1.3.1.2

Evaluate the limit of

81

$81$ which is constant as

x

$x$ approaches

81

$81$ .

\frac{0}{lim x \to 81 x - 1 \cdot 81}

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

\frac{0}{lim x \to 81 x - 1 \cdot 81}

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

Step 1.1.3.2

Evaluate the limit of

x

$x$ by plugging in

81

$81$ for

x

$x$ .

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply

- 1

$- 1$ by

81

$81$ .

\frac{0}{81 - 81}

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

Step 1.1.3.3.2

Subtract

81

$81$ from

81

$81$ .

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.2

Since

\frac{0}{0}

$\frac{0}{0}$ 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{\sqrt{x} - 9}{x - 81} = lim x \to 81 \frac{\frac{d}{d x} [\sqrt{x} - 9]}{\frac{d}{d x} [x - 81]}

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

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} [\sqrt{x} - 9]}{\frac{d}{d x} [x - 81]}

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

Step 1.3.2

By the Sum Rule, the derivative of

\sqrt{x} - 9

$\sqrt{x} - 9$ with respect to

x

$x$ is

\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 9]

$\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 9]$ .

lim x \to 81 \frac{\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x - 81]}

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

Step 1.3.3

Evaluate

\frac{d}{d x} [\sqrt{x}]

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

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Step 1.3.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

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

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

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

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

Step 1.3.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

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

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

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

Step 1.3.3.3

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.3.3.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.3.3.6.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

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

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

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

Step 1.3.4

Since

- 9

$- 9$ is constant with respect to

x

$x$ , the derivative of

- 9

$- 9$ with respect to

x

$x$ is

0

$0$ .

lim x \to 81 \frac{\frac{1}{2} x^{- \frac{1}{2}} + 0}{\frac{d}{d x} [x - 81]}

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

lim x \to 81 \frac{\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x - 81]}

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

Step 1.3.5.2

Combine terms.

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Step 1.3.5.2.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [x - 81]}

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

Step 1.3.5.2.2

Add

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

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

0

$0$ .

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 81]}

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

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 81]}

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

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 81]}

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

Step 1.3.6

By the Sum Rule, the derivative of

x - 81

$x - 81$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 81]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 81]$ .

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 81]}

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

Step 1.3.7

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 81]}

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

Step 1.3.8

Since

- 81

$- 81$ is constant with respect to

x

$x$ , the derivative of

- 81

$- 81$ with respect to

x

$x$ is

0

$0$ .

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1 + 0}

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

Step 1.3.9

Add

1

$1$ and

0

$0$ .

lim x \to 81 \frac{\frac{1}{2 x^{\frac{1}{2}}}}{1}

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 81} \frac{1}{2 x^{\frac{1}{2}}} \cdot 1$

Step 1.5

Rewrite

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

$\sqrt{x}$ .

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

Step 1.6

Multiply

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

$1$ .

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

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

$x$ .

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$81$ .

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

Step 2.3

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$81$ .

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

Step 2.4

Move the limit under the radical sign.

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

Step 3

Evaluate the limit of

$x$ by plugging in

$81$ for

$x$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{81}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Rewrite

$81$ as

$9^{2}$ .

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

Step 4.1.2

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

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

Step 4.2

Multiply

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

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Step 4.2.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{9}$ .

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

Step 4.2.2

Multiply

$2$ by

$9$ .

$\frac{1}{18}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{18}$

Decimal Form:

$0.0\overline{5}$