Calculus Examples

$\frac{lim_{x \to 0} \sqrt[3]{8 + x} - 2}{x}$

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 $0$ .

$\frac{lim_{x \to 0} \sqrt[3]{8 + x} - lim_{x \to 0} 2}{x}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to 0} 8 + x} - lim_{x \to 0} 2}{x}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt[3]{lim_{x \to 0} 8 + lim_{x \to 0} x} - lim_{x \to 0} 2}{x}$

Step 1.4

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt[3]{8 + lim_{x \to 0} x} - lim_{x \to 0} 2}{x}$

Step 1.5

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt[3]{8 + lim_{x \to 0} x} - 1 \cdot 2}{x}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt[3]{8 + 0} - 1 \cdot 2}{x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $8$ and $0$ .

$\frac{\sqrt[3]{8} - 1 \cdot 2}{x}$

Step 3.1.2

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{2^{3}} - 1 \cdot 2}{x}$

Step 3.1.3

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

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

Step 3.1.4

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{x}$

Step 3.1.5

Subtract $2$ from $2$ .

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$