Calculus Examples

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

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Multiply $- 1$ by $4$ .

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

Step 1.2.3.2

Subtract $4$ from $4$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

Move the limit under the radical sign.

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

Step 1.3.1.3

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$\frac{0}{\sqrt{lim_{x \to 4} x} - 1 \cdot 2}$

Step 1.3.2

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

$\frac{0}{\sqrt{4} - 1 \cdot 2}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

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

$\frac{0}{\sqrt{2^{2}} - 1 \cdot 2}$

Step 1.3.3.1.2

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

$\frac{0}{2 - 1 \cdot 2}$

Step 1.3.3.1.3

Multiply $- 1$ by $2$ .

$\frac{0}{2 - 2}$

Step 1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 + 0}{\frac{d}{d x} [\sqrt{x} - 2]}$

Step 3.5

Add $1$ and $0$ .

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

Step 3.6

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

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

Combine the numerators over the common denominator.

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

Step 3.7.6

Simplify the numerator.

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Step 3.7.6.1

Multiply $- 1$ by $2$ .

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

Step 3.7.6.2

Subtract $2$ from $1$ .

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

Step 3.7.7

Move the negative in front of the fraction.

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

Step 3.8

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.9.2

Combine terms.

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Step 3.9.2.1

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

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

Step 3.9.2.2

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

$lim_{x \to 4} 1 (2 \sqrt{x})$

Step 6

Evaluate the limit.

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Step 6.1

Multiply $2$ by $1$ .

$lim_{x \to 4} 2 \sqrt{x}$

Step 6.2

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

$2 lim_{x \to 4} \sqrt{x}$

Step 6.3

Move the limit under the radical sign.

$2 \sqrt{lim_{x \to 4} x}$

Step 7

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

$2 \sqrt{4}$

Step 8

Simplify the answer.

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Step 8.1

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

$2 \sqrt{2^{2}}$

Step 8.2

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

$2 \cdot 2$

Step 8.3

Multiply $2$ by $2$ .

$4$