Calculus Examples

$lim_{x \to 9} \frac{2 x^{2} - 162}{x + 3 \sqrt{x} - 18}$

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.

$\frac{lim_{x \to 9} 2 x^{2} - 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

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 $9$ .

$\frac{lim_{x \to 9} 2 x^{2} - lim_{x \to 9} 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.1.2

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

$\frac{2 lim_{x \to 9} x^{2} - lim_{x \to 9} 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.1.3

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

$\frac{2 {(lim_{x \to 9} x)}^{2} - lim_{x \to 9} 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.1.4

Evaluate the limit of $162$ which is constant as $x$ approaches $9$ .

$\frac{2 {(lim_{x \to 9} x)}^{2} - 1 \cdot 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.2

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

$\frac{2 \cdot 9^{2} - 1 \cdot 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

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

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

$\frac{2 \cdot 81 - 1 \cdot 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.3.1.2

Multiply $2$ by $81$ .

$\frac{162 - 1 \cdot 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.3.1.3

Multiply $- 1$ by $162$ .

$\frac{162 - 162}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

Step 1.1.2.3.2

Subtract $162$ from $162$ .

$\frac{0}{lim_{x \to 9} x + 3 \sqrt{x} - 18}$

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 $9$ .

$\frac{0}{lim_{x \to 9} x + lim_{x \to 9} 3 \sqrt{x} - lim_{x \to 9} 18}$

Step 1.1.3.2

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

$\frac{0}{lim_{x \to 9} x + 3 lim_{x \to 9} \sqrt{x} - lim_{x \to 9} 18}$

Step 1.1.3.3

Move the limit under the radical sign.

$\frac{0}{lim_{x \to 9} x + 3 \sqrt{lim_{x \to 9} x} - lim_{x \to 9} 18}$

Step 1.1.3.4

Evaluate the limit of $18$ which is constant as $x$ approaches $9$ .

$\frac{0}{lim_{x \to 9} x + 3 \sqrt{lim_{x \to 9} x} - 1 \cdot 18}$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$\frac{0}{9 + 3 \sqrt{lim_{x \to 9} x} - 1 \cdot 18}$

Step 1.1.3.5.2

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

$\frac{0}{9 + 3 \sqrt{9} - 1 \cdot 18}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

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

$\frac{0}{9 + 3 \sqrt{3^{2}} - 1 \cdot 18}$

Step 1.1.3.6.1.2

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

$\frac{0}{9 + 3 \cdot 3 - 1 \cdot 18}$

Step 1.1.3.6.1.3

Multiply $3$ by $3$ .

$\frac{0}{9 + 9 - 1 \cdot 18}$

Step 1.1.3.6.1.4

Multiply $- 1$ by $18$ .

$\frac{0}{9 + 9 - 18}$

Step 1.1.3.6.2

Add $9$ and $9$ .

$\frac{0}{18 - 18}$

Step 1.1.3.6.3

Subtract $18$ from $18$ .

$\frac{0}{0}$

Step 1.1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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.

$lim_{x \to 9} \frac{2 x^{2} - 162}{x + 3 \sqrt{x} - 18} = lim_{x \to 9} \frac{\frac{d}{d x} [2 x^{2} - 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 9} \frac{\frac{d}{d x} [2 x^{2} - 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 x^{2} - 162$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 162]$ .

$lim_{x \to 9} \frac{\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.3

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

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Step 1.3.3.1

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

$lim_{x \to 9} \frac{2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.3.2

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

$lim_{x \to 9} \frac{2 (2 x) + \frac{d}{d x} [- 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.3.3

Multiply $2$ by $2$ .

$lim_{x \to 9} \frac{4 x + \frac{d}{d x} [- 162]}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.4

Since $- 162$ is constant with respect to $x$ , the derivative of $- 162$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{4 x + 0}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.5

Add $4 x$ and $0$ .

$lim_{x \to 9} \frac{4 x}{\frac{d}{d x} [x + 3 \sqrt{x} - 18]}$

Step 1.3.6

By the Sum Rule, the derivative of $x + 3 \sqrt{x} - 18$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [- 18]$ .

$lim_{x \to 9} \frac{4 x}{\frac{d}{d x} [x] + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [- 18]}$

Step 1.3.7

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

$lim_{x \to 9} \frac{4 x}{1 + \frac{d}{d x} [3 \sqrt{x}] + \frac{d}{d x} [- 18]}$

Step 1.3.8

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

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Step 1.3.8.1

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

$lim_{x \to 9} \frac{4 x}{1 + \frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [- 18]}$

Step 1.3.8.2

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

$lim_{x \to 9} \frac{4 x}{1 + 3 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 18]}$

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

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.4

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

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.5

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

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.6

Combine the numerators over the common denominator.

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.7

Simplify the numerator.

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Step 1.3.8.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.7.2

Subtract $2$ from $1$ .

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.8

Move the negative in front of the fraction.

$lim_{x \to 9} \frac{4 x}{1 + 3 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 18]}$

Step 1.3.8.9

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

$lim_{x \to 9} \frac{4 x}{1 + 3 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 18]}$

Step 1.3.8.10

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

$lim_{x \to 9} \frac{4 x}{1 + \frac{3 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 18]}$

Step 1.3.8.11

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

$lim_{x \to 9} \frac{4 x}{1 + \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 18]}$

Step 1.3.9

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{4 x}{1 + \frac{3}{2 x^{\frac{1}{2}}} + 0}$

Step 1.3.10

Add $1 + \frac{3}{2 x^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 9} \frac{4 x}{1 + \frac{3}{2 x^{\frac{1}{2}}}}$

Step 1.4

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

$lim_{x \to 9} \frac{4 x}{1 + \frac{3}{2 \sqrt{x}}}$

Step 1.5

Combine terms.

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Step 1.5.1

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

$lim_{x \to 9} \frac{4 x}{\frac{2 \sqrt{x}}{2 \sqrt{x}} + \frac{3}{2 \sqrt{x}}}$

Step 1.5.2

Combine the numerators over the common denominator.

$lim_{x \to 9} \frac{4 x}{\frac{2 \sqrt{x} + 3}{2 \sqrt{x}}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to 9} \frac{x}{\frac{2 \sqrt{x} + 3}{2 \sqrt{x}}}$

Step 2.2

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

$4 \frac{lim_{x \to 9} x}{lim_{x \to 9} \frac{2 \sqrt{x} + 3}{2 \sqrt{x}}}$

Step 2.3

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

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} lim_{x \to 9} \frac{2 \sqrt{x} + 3}{\sqrt{x}}}$

Step 2.4

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

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{lim_{x \to 9} 2 \sqrt{x} + 3}{lim_{x \to 9} \sqrt{x}}}$

Step 2.5

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

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{lim_{x \to 9} 2 \sqrt{x} + lim_{x \to 9} 3}{lim_{x \to 9} \sqrt{x}}}$

Step 2.6

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

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{2 lim_{x \to 9} \sqrt{x} + lim_{x \to 9} 3}{lim_{x \to 9} \sqrt{x}}}$

Step 2.7

Move the limit under the radical sign.

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 9} x} + lim_{x \to 9} 3}{lim_{x \to 9} \sqrt{x}}}$

Step 2.8

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

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 9} x} + 3}{lim_{x \to 9} \sqrt{x}}}$

Step 2.9

Move the limit under the radical sign.

$4 \frac{lim_{x \to 9} x}{\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 9} x} + 3}{\sqrt{lim_{x \to 9} x}}}$

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 $x$ .

$4 \frac{9}{\frac{1}{2} \cdot \frac{2 \sqrt{lim_{x \to 9} x} + 3}{\sqrt{lim_{x \to 9} x}}}$

Step 3.2

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{2 \sqrt{9} + 3}{\sqrt{lim_{x \to 9} x}}}$

Step 3.3

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{2 \sqrt{9} + 3}{\sqrt{9}}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{2 \sqrt{3^{2}} + 3}{\sqrt{9}}}$

Step 4.1.2

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{2 \cdot 3 + 3}{\sqrt{9}}}$

Step 4.1.3

Multiply $2$ by $3$ .

$4 \frac{9}{\frac{1}{2} \cdot \frac{6 + 3}{\sqrt{9}}}$

Step 4.1.4

Add $6$ and $3$ .

$4 \frac{9}{\frac{1}{2} \cdot \frac{9}{\sqrt{9}}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{9}{\sqrt{3^{2}}}}$

Step 4.2.2

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

$4 \frac{9}{\frac{1}{2} \cdot \frac{9}{3}}$

Step 4.3

Multiply $\frac{1}{2}$ by $\frac{9}{3}$ .

$4 \frac{9}{\frac{9}{2 \cdot 3}}$

Step 4.4

Multiply $2$ by $3$ .

$4 \frac{9}{\frac{9}{6}}$

Step 4.5

Cancel the common factor of $9$ and $6$ .

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Step 4.5.1

Factor $3$ out of $9$ .

$4 \frac{9}{\frac{3 (3)}{6}}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $3$ out of $6$ .

$4 \frac{9}{\frac{3 \cdot 3}{3 \cdot 2}}$

Step 4.5.2.2

Cancel the common factor.

$4 \frac{9}{\frac{3 \cdot 3}{3 \cdot 2}}$

Step 4.5.2.3

Rewrite the expression.

$4 \frac{9}{\frac{3}{2}}$

Step 4.6

Multiply the numerator by the reciprocal of the denominator.

$4 (9 (\frac{2}{3}))$

Step 4.7

Cancel the common factor of $3$ .

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Step 4.7.1

Factor $3$ out of $9$ .

$4 (3 (3) \frac{2}{3})$

Step 4.7.2

Cancel the common factor.

$4 (3 \cdot 3 \frac{2}{3})$

Step 4.7.3

Rewrite the expression.

$4 (3 \cdot 2)$

Step 4.8

Multiply $3$ by $2$ .

$4 \cdot 6$

Step 4.9

Multiply $4$ by $6$ .

$24$