Calculus Examples

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

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.

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

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 $y$ approaches $4$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

$\frac{{(lim_{y \to 4} y)}^{2} - 1 \cdot 16}{lim_{y \to 4} \sqrt{y} - 2}$

Step 1.1.2.2

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

$\frac{4^{2} - 1 \cdot 16}{lim_{y \to 4} \sqrt{y} - 2}$

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 $2$ .

$\frac{16 - 1 \cdot 16}{lim_{y \to 4} \sqrt{y} - 2}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $16$ .

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

Step 1.1.2.3.2

Subtract $16$ from $16$ .

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

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 $y$ approaches $4$ .

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

Step 1.1.3.1.2

Move the limit under the radical sign.

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

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

Step 1.1.3.3.1.2

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

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

Step 1.1.3.3.1.3

Multiply $- 1$ by $2$ .

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

Step 1.1.3.3.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $2 y$ and $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

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

Step 1.3.7.3

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

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

Step 1.3.7.4

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

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

Step 1.3.7.5

Combine the numerators over the common denominator.

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

Step 1.3.7.6

Simplify the numerator.

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Step 1.3.7.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.7.6.2

Subtract $2$ from $1$ .

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

Step 1.3.7.7

Move the negative in front of the fraction.

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

Step 1.3.8

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

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

Step 1.3.9

Simplify.

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Step 1.3.9.1

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

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

Step 1.3.9.2

Combine terms.

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Step 1.3.9.2.1

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

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

Step 1.3.9.2.2

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

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

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

$lim_{y \to 4} 2 y (2 \sqrt{y})$

Step 1.6

Multiply $2$ by $2$ .

$lim_{y \to 4} 4 y \sqrt{y}$

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 $y$ .

$4 lim_{y \to 4} y \sqrt{y}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $y$ approaches $4$ .

$4 lim_{y \to 4} y \cdot lim_{y \to 4} \sqrt{y}$

Step 2.3

Move the limit under the radical sign.

$4 lim_{y \to 4} y \cdot \sqrt{lim_{y \to 4} y}$

Step 3

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

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Step 3.1

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

$4 \cdot 4 \cdot \sqrt{lim_{y \to 4} y}$

Step 3.2

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

$4 \cdot 4 \cdot \sqrt{4}$

Step 4

Simplify the answer.

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Step 4.1

Multiply $4$ by $4$ .

$16 \cdot \sqrt{4}$

Step 4.2

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

$16 \cdot \sqrt{2^{2}}$

Step 4.3

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

$16 \cdot 2$

Step 4.4

Multiply $16$ by $2$ .

$32$