Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Move the limit under the radical sign.

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

Step 1.1.2.2

Move the exponent $2$ from ${(5 - x)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Simplify terms.

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Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

Simplify the answer.

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Step 1.1.2.5.2.1

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

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

Step 1.1.2.5.2.2

Subtract $5$ from $5$ .

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

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 $5$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

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

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

Multiply $- 1$ by $1$ .

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

Step 1.3.9

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

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

Step 1.3.10

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

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

Step 1.3.11

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

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Step 1.3.11.1

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

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

Step 1.3.11.2

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

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

Step 1.3.11.3

Multiply $- 1$ by $1$ .

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

Step 1.3.12

Subtract $1$ from $0$ .

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

Step 1.4

Reduce.

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Step 1.4.1

Dividing two negative values results in a positive value.

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

Step 1.4.2

Cancel the common factor of $1$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

$lim_{x \to 5} 1$

Step 2

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

$1$