Calculus Examples

$\frac{lim_{x \to 8} (\sqrt{x} + \sqrt[3]{8 x}) + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} \sqrt{x} + lim_{x \to 8} \sqrt[3]{8 x} + lim_{x \to 8} 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x} + lim_{x \to 8} \sqrt[3]{8 x} + lim_{x \to 8} 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x} + \sqrt[3]{lim_{x \to 8} 8 x} + lim_{x \to 8} 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{lim_{x \to 8} x} + \sqrt[3]{8 lim_{x \to 8} x} + lim_{x \to 8} 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} x} + \sqrt[3]{8 lim_{x \to 8} x} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{8} + \sqrt[3]{8 lim_{x \to 8} x} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{8} + \sqrt[3]{8 \cdot 8} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)} + \sqrt[3]{8 \cdot 8} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.1.2

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

$\frac{\sqrt{2^{2} \cdot 2} + \sqrt[3]{8 \cdot 8} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2} + \sqrt[3]{8 \cdot 8} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.3

Multiply $8$ by $8$ .

$\frac{2 \sqrt{2} + \sqrt[3]{64} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.4

Rewrite $64$ as $4^{3}$ .

$\frac{2 \sqrt{2} + \sqrt[3]{4^{3}} + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.5

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

$\frac{2 \sqrt{2} + 4 + 4}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.1.6

Add $4$ and $4$ .

$\frac{2 \sqrt{2} + 8}{\sqrt{4 x} + \sqrt[3]{x} + 4}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

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

$\frac{2 \sqrt{2} + 8}{\sqrt{2^{2} x} + \sqrt[3]{x} + 4}$

Step 7.2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2} + 8}{2 \sqrt{x} + \sqrt[3]{x} + 4}$