Calculus Examples

$lim_{x \to 4} \frac{16 - x^{2}}{\sqrt{4 x} - x}$

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.

$\frac{lim_{x \to 4} 16 - x^{2}}{lim_{x \to 4} \sqrt{4 x} - x}$

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 $4$ .

$\frac{lim_{x \to 4} 16 - lim_{x \to 4} x^{2}}{lim_{x \to 4} \sqrt{4 x} - x}$

Step 1.1.2.1.2

Evaluate the limit of $16$ which is constant as $x$ approaches $4$ .

$\frac{16 - lim_{x \to 4} x^{2}}{lim_{x \to 4} \sqrt{4 x} - x}$

Step 1.1.2.1.3

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

$\frac{16 - {(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} \sqrt{4 x} - x}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{16 - 4^{2}}{lim_{x \to 4} \sqrt{4 x} - x}$

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 $4$ to the power of $2$ .

$\frac{16 - 1 \cdot 16}{lim_{x \to 4} \sqrt{4 x} - x}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $16$ .

$\frac{16 - 16}{lim_{x \to 4} \sqrt{4 x} - x}$

Step 1.1.2.3.2

Subtract $16$ from $16$ .

$\frac{0}{lim_{x \to 4} \sqrt{4 x} - x}$

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 $4$ .

$\frac{0}{lim_{x \to 4} \sqrt{4 x} - lim_{x \to 4} x}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 4} 4 x} - lim_{x \to 4} x}$

Step 1.1.3.3

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

$\frac{0}{\sqrt{4 lim_{x \to 4} x} - lim_{x \to 4} x}$

Step 1.1.3.4

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 1.1.3.4.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{\sqrt{4 \cdot 4} - lim_{x \to 4} x}$

Step 1.1.3.4.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{\sqrt{4 \cdot 4} - 4}$

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 $4$ by $4$ .

$\frac{0}{\sqrt{16} - 4}$

Step 1.1.3.5.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{0}{\sqrt{4^{2}} - 4}$

Step 1.1.3.5.1.3

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

$\frac{0}{4 - 4}$

Step 1.1.3.5.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

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.

$lim_{x \to 4} \frac{16 - x^{2}}{\sqrt{4 x} - x} = lim_{x \to 4} \frac{\frac{d}{d x} [16 - x^{2}]}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [16 - x^{2}]}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.2

By the Sum Rule, the derivative of $16 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [16] + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.3

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{0 + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 4} \frac{0 - \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.4.2

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

$lim_{x \to 4} \frac{0 - (2 x)}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.4.3

Multiply $2$ by $- 1$ .

$lim_{x \to 4} \frac{0 - 2 x}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.5

Subtract $2 x$ from $0$ .

$lim_{x \to 4} \frac{- 2 x}{\frac{d}{d x} [\sqrt{4 x} - x]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{4 x} - x$ with respect to $x$ is $\frac{d}{d x} [\sqrt{4 x}] + \frac{d}{d x} [- x]$ .

$lim_{x \to 4} \frac{- 2 x}{\frac{d}{d x} [\sqrt{4 x}] + \frac{d}{d x} [- x]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [\sqrt{4 x}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x}$ as ${(4 x)}^{\frac{1}{2}}$ .

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

Step 1.3.7.2

Factor $4$ out of $4 x$ .

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

Step 1.3.7.3

Apply the product rule to $4 (x)$ .

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

Step 1.3.7.4

Rewrite $4$ as $2^{2}$ .

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

Step 1.3.7.5

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.3.7.6

Cancel the common factor of $2$ .

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Step 1.3.7.6.1

Cancel the common factor.

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

Step 1.3.7.6.2

Rewrite the expression.

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

Step 1.3.7.7

Evaluate the exponent.

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

Step 1.3.7.8

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

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

Step 1.3.7.9

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

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

Step 1.3.7.10

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

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

Step 1.3.7.11

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

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

Step 1.3.7.12

Combine the numerators over the common denominator.

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

Step 1.3.7.13

Simplify the numerator.

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Step 1.3.7.13.1

Multiply $- 1$ by $2$ .

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

Step 1.3.7.13.2

Subtract $2$ from $1$ .

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

Step 1.3.7.14

Move the negative in front of the fraction.

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

Step 1.3.7.15

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

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

Step 1.3.7.16

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

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

Step 1.3.7.17

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

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

Step 1.3.7.18

Cancel the common factor.

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

Step 1.3.7.19

Rewrite the expression.

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

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 $n = 1$ .

$lim_{x \to 4} \frac{- 2 x}{\frac{1}{x^{\frac{1}{2}}} - 1 \cdot 1}$

Step 1.3.8.3

Multiply $- 1$ by $1$ .

$lim_{x \to 4} \frac{- 2 x}{\frac{1}{x^{\frac{1}{2}}} - 1}$

Step 1.4

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

$lim_{x \to 4} \frac{- 2 x}{\frac{1}{\sqrt{x}} - 1}$

Step 1.5

Combine terms.

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Step 1.5.1

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

$lim_{x \to 4} \frac{- 2 x}{\frac{1}{\sqrt{x}} - 1 \cdot \frac{\sqrt{x}}{\sqrt{x}}}$

Step 1.5.2

Combine $- 1$ and $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.5.3

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{- 2 x}{\frac{1 - \sqrt{x}}{\sqrt{x}}}$

Step 2

Evaluate the limit.

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Step 2.1

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

$- 2 lim_{x \to 4} \frac{x}{\frac{1 - \sqrt{x}}{\sqrt{x}}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$- 2 \frac{lim_{x \to 4} x}{lim_{x \to 4} \frac{1 - \sqrt{x}}{\sqrt{x}}}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$- 2 \frac{lim_{x \to 4} x}{\frac{lim_{x \to 4} 1 - \sqrt{x}}{lim_{x \to 4} \sqrt{x}}}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$- 2 \frac{lim_{x \to 4} x}{\frac{lim_{x \to 4} 1 - lim_{x \to 4} \sqrt{x}}{lim_{x \to 4} \sqrt{x}}}$

Step 2.5

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$- 2 \frac{lim_{x \to 4} x}{\frac{1 - lim_{x \to 4} \sqrt{x}}{lim_{x \to 4} \sqrt{x}}}$

Step 2.6

Move the limit under the radical sign.

$- 2 \frac{lim_{x \to 4} x}{\frac{1 - \sqrt{lim_{x \to 4} x}}{lim_{x \to 4} \sqrt{x}}}$

Step 2.7

Move the limit under the radical sign.

$- 2 \frac{lim_{x \to 4} x}{\frac{1 - \sqrt{lim_{x \to 4} x}}{\sqrt{lim_{x \to 4} x}}}$

Step 3

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 2 \frac{4}{\frac{1 - \sqrt{lim_{x \to 4} x}}{\sqrt{lim_{x \to 4} x}}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 2 \frac{4}{\frac{1 - \sqrt{4}}{\sqrt{lim_{x \to 4} x}}}$

Step 3.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 2 \frac{4}{\frac{1 - \sqrt{4}}{\sqrt{4}}}$

Step 4

Simplify the answer.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$- 2 (4 \frac{\sqrt{4}}{1 - \sqrt{4}})$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $4$ as $2^{2}$ .

$- 2 (4 \frac{\sqrt{2^{2}}}{1 - \sqrt{4}})$

Step 4.2.2

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

$- 2 (4 \frac{2}{1 - \sqrt{4}})$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $4$ as $2^{2}$ .

$- 2 (4 \frac{2}{1 - \sqrt{2^{2}}})$

Step 4.3.2

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

$- 2 (4 \frac{2}{1 - 1 \cdot 2})$

Step 4.3.3

Multiply $- 1$ by $2$ .

$- 2 (4 \frac{2}{1 - 2})$

Step 4.3.4

Subtract $2$ from $1$ .

$- 2 (4 (\frac{2}{- 1}))$

Step 4.4

Divide $2$ by $- 1$ .

$- 2 (4 \cdot - 2)$

Step 4.5

Multiply $- 2 (4 \cdot - 2)$ .

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Step 4.5.1

Multiply $4$ by $- 2$ .

$- 2 \cdot - 8$

Step 4.5.2

Multiply $- 2$ by $- 8$ .

$16$