Calculus Examples

$lim_{x \to 8} \frac{x - \sqrt[3]{1 - x^{3}}}{1}$

Step 1

Divide $x - \sqrt[3]{1 - x^{3}}$ by $1$ .

$lim_{x \to 8} x - \sqrt[3]{1 - x^{3}}$

Step 2

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

$lim_{x \to 8} x - lim_{x \to 8} \sqrt[3]{1 - x^{3}}$

Step 3

Move the limit under the radical sign.

$lim_{x \to 8} x - \sqrt[3]{lim_{x \to 8} 1 - x^{3}}$

Step 4

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

$lim_{x \to 8} x - \sqrt[3]{lim_{x \to 8} 1 - lim_{x \to 8} x^{3}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$lim_{x \to 8} x - \sqrt[3]{1 - lim_{x \to 8} x^{3}}$

Step 6

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

$lim_{x \to 8} x - \sqrt[3]{1 - {(lim_{x \to 8} x)}^{3}}$

Step 7

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

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Step 7.1

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

$8 - \sqrt[3]{1 - {(lim_{x \to 8} x)}^{3}}$

Step 7.2

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

$8 - \sqrt[3]{1 - 8^{3}}$

Step 8

Simplify each term.

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Step 8.1

Raise $8$ to the power of $3$ .

$8 - \sqrt[3]{1 - 1 \cdot 512}$

Step 8.2

Multiply $- 1$ by $512$ .

$8 - \sqrt[3]{1 - 512}$

Step 8.3

Subtract $512$ from $1$ .

$8 - \sqrt[3]{- 511}$

Step 8.4

Rewrite $- 511$ as ${(- 1)}^{3} \cdot 511$ .

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Step 8.4.1

Rewrite $- 511$ as $- 1 (511)$ .

$8 - \sqrt[3]{- 1 (511)}$

Step 8.4.2

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

$8 - \sqrt[3]{{(- 1)}^{3} \cdot 511}$

Step 8.5

Pull terms out from under the radical.

$8 - (- 1 \sqrt[3]{511})$

Step 8.6

Rewrite $- 1 \sqrt[3]{511}$ as $- \sqrt[3]{511}$ .

$8 - - \sqrt[3]{511}$

Step 8.7

Multiply $- - \sqrt[3]{511}$ .

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Step 8.7.1

Multiply $- 1$ by $- 1$ .

$8 + 1 \sqrt[3]{511}$

Step 8.7.2

Multiply $\sqrt[3]{511}$ by $1$ .

$8 + \sqrt[3]{511}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$8 + \sqrt[3]{511}$

Decimal Form:

$15.99478827 \dots$