Calculus Examples

$lim_{x \to - 1} \sqrt[3]{x^{3} - 1}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to - 1} x^{3} - 1}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\sqrt[3]{lim_{x \to - 1} x^{3} - lim_{x \to - 1} 1}$

Step 1.3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\sqrt[3]{{(lim_{x \to - 1} x)}^{3} - lim_{x \to - 1} 1}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- 1$ .

$\sqrt[3]{{(lim_{x \to - 1} x)}^{3} - 1 \cdot 1}$

Step 2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\sqrt[3]{{(- 1)}^{3} - 1 \cdot 1}$

Step 3

Simplify the answer.

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Step 3.1

Raise $- 1$ to the power of $3$ .

$\sqrt[3]{- 1 - 1 \cdot 1}$

Step 3.2

Multiply $- 1$ by $1$ .

$\sqrt[3]{- 1 - 1}$

Step 3.3

Subtract $1$ from $- 1$ .

$\sqrt[3]{- 2}$

Step 3.4

Rewrite $- 2$ as ${(- 1)}^{3} \cdot 2$ .

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Step 3.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$\sqrt[3]{- 1 (2)}$

Step 3.4.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 2}$

Step 3.5

Pull terms out from under the radical.

$- 1 \sqrt[3]{2}$

Step 3.6

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$- \sqrt[3]{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[3]{2}$

Decimal Form:

$- 1.25992104 \dots$