Precalculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Subtract $8$ from $512$ .

$\sqrt[3]{504} - 8 - 1 \cdot 1$

Step 7.1.3

Rewrite $504$ as $2^{3} \cdot 63$ .

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Step 7.1.3.1

Factor $8$ out of $504$ .

$\sqrt[3]{8 (63)} - 8 - 1 \cdot 1$

Step 7.1.3.2

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

$\sqrt[3]{2^{3} \cdot 63} - 8 - 1 \cdot 1$

Step 7.1.4

Pull terms out from under the radical.

$2 \sqrt[3]{63} - 8 - 1 \cdot 1$

Step 7.1.5

Multiply $- 1$ by $1$ .

$2 \sqrt[3]{63} - 8 - 1$

Step 7.2

Subtract $1$ from $- 8$ .

$2 \sqrt[3]{63} - 9$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[3]{63} - 9$

Decimal Form:

$- 1.04188558 \dots$