Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $16$ .

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

Step 1.2

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

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

Step 2

Evaluate the limit of $x$ by plugging in $16$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $4$ .

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

Step 3.1.2

Subtract $4$ from $16$ .

$\frac{12}{\sqrt{x} - 16}$

Step 3.2

Multiply $\frac{12}{\sqrt{x} - 16}$ by $\frac{\sqrt{x} + 16}{\sqrt{x} + 16}$ .

$\frac{12}{\sqrt{x} - 16} \cdot \frac{\sqrt{x} + 16}{\sqrt{x} + 16}$

Step 3.3

Multiply $\frac{12}{\sqrt{x} - 16}$ by $\frac{\sqrt{x} + 16}{\sqrt{x} + 16}$ .

$\frac{12 (\sqrt{x} + 16)}{(\sqrt{x} - 16) (\sqrt{x} + 16)}$

Step 3.4

Expand the denominator using the FOIL method.

$\frac{12 (\sqrt{x} + 16)}{{\sqrt{x}}^{2} + \sqrt{x} \cdot 16 - 16 \sqrt{x} - 256}$

Step 3.5

Simplify.

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Step 3.5.1

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

$\frac{12 (\sqrt{x} + 16)}{{(x^{\frac{1}{2}})}^{2} - 256}$

Step 3.5.2

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

$\frac{12 (\sqrt{x} + 16)}{x^{\frac{1}{2} \cdot 2} - 256}$

Step 3.5.3

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

$\frac{12 (\sqrt{x} + 16)}{x^{\frac{2}{2}} - 256}$

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{12 (\sqrt{x} + 16)}{x^{\frac{2}{2}} - 256}$

Step 3.5.4.2

Rewrite the expression.

$\frac{12 (\sqrt{x} + 16)}{x^{1} - 256}$

Step 3.5.5

Simplify.

$\frac{12 (\sqrt{x} + 16)}{x - 256}$