Calculus Examples

$lim_{x \to 5} \frac{5 - \sqrt{5 x}}{x - 5}$

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.

$\frac{lim_{x \to 5} 5 - \sqrt{5 x}}{lim_{x \to 5} x - 5}$

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 $5$ .

$\frac{lim_{x \to 5} 5 - lim_{x \to 5} \sqrt{5 x}}{lim_{x \to 5} x - 5}$

Step 1.1.2.1.2

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{5 - lim_{x \to 5} \sqrt{5 x}}{lim_{x \to 5} x - 5}$

Step 1.1.2.1.3

Move the limit under the radical sign.

$\frac{5 - \sqrt{lim_{x \to 5} 5 x}}{lim_{x \to 5} x - 5}$

Step 1.1.2.1.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 - \sqrt{5 lim_{x \to 5} x}}{lim_{x \to 5} x - 5}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5 - \sqrt{5 \cdot 5}}{lim_{x \to 5} x - 5}$

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

Multiply $5$ by $5$ .

$\frac{5 - \sqrt{25}}{lim_{x \to 5} x - 5}$

Step 1.1.2.3.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{5 - \sqrt{5^{2}}}{lim_{x \to 5} x - 5}$

Step 1.1.2.3.1.3

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

$\frac{5 - 1 \cdot 5}{lim_{x \to 5} x - 5}$

Step 1.1.2.3.1.4

Multiply $- 1$ by $5$ .

$\frac{5 - 5}{lim_{x \to 5} x - 5}$

Step 1.1.2.3.2

Subtract $5$ from $5$ .

$\frac{0}{lim_{x \to 5} x - 5}$

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 $5$ .

$\frac{0}{lim_{x \to 5} x - lim_{x \to 5} 5}$

Step 1.1.3.1.2

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{0}{lim_{x \to 5} x - 1 \cdot 5}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{0}{5 - 1 \cdot 5}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $5$ .

$\frac{0}{5 - 5}$

Step 1.1.3.3.2

Subtract $5$ from $5$ .

$\frac{0}{0}$

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.

$lim_{x \to 5} \frac{5 - \sqrt{5 x}}{x - 5} = lim_{x \to 5} \frac{\frac{d}{d x} [5 - \sqrt{5 x}]}{\frac{d}{d x} [x - 5]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 5} \frac{\frac{d}{d x} [5 - \sqrt{5 x}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.2

By the Sum Rule, the derivative of $5 - \sqrt{5 x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- \sqrt{5 x}]$ .

$lim_{x \to 5} \frac{\frac{d}{d x} [5] + \frac{d}{d x} [- \sqrt{5 x}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{0 + \frac{d}{d x} [- \sqrt{5 x}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \sqrt{5 x}]$ .

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Step 1.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$lim_{x \to 5} \frac{0 + \frac{d}{d x} [- {(5 x)}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.2

Factor $5$ out of $5 x$ .

$lim_{x \to 5} \frac{0 + \frac{d}{d x} [- {(5 (x))}^{\frac{1}{2}}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.3

Apply the product rule to $5 (x)$ .

$lim_{x \to 5} \frac{0 + \frac{d}{d x} [- (5^{\frac{1}{2}} x^{\frac{1}{2}})]}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.4

Since $- 5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $- 5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $- 5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.5

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

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.6

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

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.7

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

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.8

Combine the numerators over the common denominator.

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.9

Simplify the numerator.

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Step 1.3.4.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.9.2

Subtract $2$ from $1$ .

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.10

Move the negative in front of the fraction.

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}})}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.11

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

$lim_{x \to 5} \frac{0 - 5^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $5^{\frac{1}{2}}$ .

$lim_{x \to 5} \frac{0 - \frac{x^{- \frac{1}{2}} \cdot 5^{\frac{1}{2}}}{2}}{\frac{d}{d x} [x - 5]}$

Step 1.3.4.13

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

$lim_{x \to 5} \frac{0 - \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 5]}$

Step 1.3.5

Subtract $\frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ from $0$ .

$lim_{x \to 5} \frac{- \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x - 5]}$

Step 1.3.6

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$lim_{x \to 5} \frac{- \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 5]}$

Step 1.3.7

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

$lim_{x \to 5} \frac{- \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{1 + \frac{d}{d x} [- 5]}$

Step 1.3.8

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{- \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 5} \frac{- \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 5} - \frac{5^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot 1$

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

Rewrite $5^{\frac{1}{2}}$ as $\sqrt{5}$ .

$lim_{x \to 5} - \frac{\sqrt{5}}{2 x^{\frac{1}{2}}} \cdot 1$

Step 1.5.2

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

$lim_{x \to 5} - \frac{\sqrt{5}}{2 \sqrt{x}} \cdot 1$

Step 1.6

Multiply $- 1$ by $1$ .

$lim_{x \to 5} - \frac{\sqrt{5}}{2 \sqrt{x}}$

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 $x$ .

$- lim_{x \to 5} \frac{\sqrt{5}}{2 \sqrt{x}}$

Step 2.2

Move the term $\frac{\sqrt{5}}{2}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{\sqrt{5}}{2} lim_{x \to 5} \frac{1}{\sqrt{x}}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$- \frac{\sqrt{5}}{2} \cdot \frac{lim_{x \to 5} 1}{lim_{x \to 5} \sqrt{x}}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$- \frac{\sqrt{5}}{2} \cdot \frac{1}{lim_{x \to 5} \sqrt{x}}$

Step 2.5

Move the limit under the radical sign.

$- \frac{\sqrt{5}}{2} \cdot \frac{1}{\sqrt{lim_{x \to 5} x}}$

Step 3

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$- \frac{\sqrt{5}}{2} \cdot \frac{1}{\sqrt{5}}$

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $\sqrt{5}$ .

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Step 4.1.1

Move the leading negative in $- \frac{\sqrt{5}}{2}$ into the numerator.

$\frac{- \sqrt{5}}{2} \cdot \frac{1}{\sqrt{5}}$

Step 4.1.2

Factor $\sqrt{5}$ out of $- \sqrt{5}$ .

$\frac{\sqrt{5} \cdot - 1}{2} \cdot \frac{1}{\sqrt{5}}$

Step 4.1.3

Cancel the common factor.

$\frac{\sqrt{5} \cdot - 1}{2} \cdot \frac{1}{\sqrt{5}}$

Step 4.1.4

Rewrite the expression.

$\frac{- 1}{2}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$