Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the answer.

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Step 10.1

Multiply $8$ by $8^{3}$ by adding the exponents.

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Step 10.1.1

Multiply $8$ by $8^{3}$ .

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Step 10.1.1.1

Raise $8$ to the power of $1$ .

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

Step 10.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.1.2

Add $1$ and $3$ .

$\sqrt[3]{\frac{8^{4} - 8}{8^{3} + 1}}$

Step 10.2

Raise $8$ to the power of $4$ .

$\sqrt[3]{\frac{4096 - 8}{8^{3} + 1}}$

Step 10.3

Subtract $8$ from $4096$ .

$\sqrt[3]{\frac{4088}{8^{3} + 1}}$

Step 10.4

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

$\sqrt[3]{\frac{4088}{512 + 1}}$

Step 10.5

Add $512$ and $1$ .

$\sqrt[3]{\frac{4088}{513}}$

Step 10.6

Rewrite $\sqrt[3]{\frac{4088}{513}}$ as $\frac{\sqrt[3]{4088}}{\sqrt[3]{513}}$ .

$\frac{\sqrt[3]{4088}}{\sqrt[3]{513}}$

Step 10.7

Simplify the numerator.

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Step 10.7.1

Rewrite $4088$ as $2^{3} \cdot 511$ .

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Step 10.7.1.1

Factor $8$ out of $4088$ .

$\frac{\sqrt[3]{8 (511)}}{\sqrt[3]{513}}$

Step 10.7.1.2

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

$\frac{\sqrt[3]{2^{3} \cdot 511}}{\sqrt[3]{513}}$

Step 10.7.2

Pull terms out from under the radical.

$\frac{2 \sqrt[3]{511}}{\sqrt[3]{513}}$

Step 10.8

Simplify the denominator.

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Step 10.8.1

Rewrite $513$ as $3^{3} \cdot 19$ .

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Step 10.8.1.1

Factor $27$ out of $513$ .

$\frac{2 \sqrt[3]{511}}{\sqrt[3]{27 (19)}}$

Step 10.8.1.2

Rewrite $27$ as $3^{3}$ .

$\frac{2 \sqrt[3]{511}}{\sqrt[3]{3^{3} \cdot 19}}$

Step 10.8.2

Pull terms out from under the radical.

$\frac{2 \sqrt[3]{511}}{3 \sqrt[3]{19}}$

Step 10.9

Multiply $\frac{2 \sqrt[3]{511}}{3 \sqrt[3]{19}}$ by $\frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$ .

$\frac{2 \sqrt[3]{511}}{3 \sqrt[3]{19}} \cdot \frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$

Step 10.10

Combine and simplify the denominator.

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Step 10.10.1

Multiply $\frac{2 \sqrt[3]{511}}{3 \sqrt[3]{19}}$ by $\frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \sqrt[3]{19} {\sqrt[3]{19}}^{2}}$

Step 10.10.2

Move $\sqrt[3]{19}$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 (\sqrt[3]{19} {\sqrt[3]{19}}^{2})}$

Step 10.10.3

Raise $\sqrt[3]{19}$ to the power of $1$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 ({\sqrt[3]{19}}^{1} {\sqrt[3]{19}}^{2})}$

Step 10.10.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 {\sqrt[3]{19}}^{1 + 2}}$

Step 10.10.5

Add $1$ and $2$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 {\sqrt[3]{19}}^{3}}$

Step 10.10.6

Rewrite ${\sqrt[3]{19}}^{3}$ as $19$ .

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Step 10.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{19}$ as $19^{\frac{1}{3}}$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 {(19^{\frac{1}{3}})}^{3}}$

Step 10.10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \cdot 19^{\frac{1}{3} \cdot 3}}$

Step 10.10.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \cdot 19^{\frac{3}{3}}}$

Step 10.10.6.4

Cancel the common factor of $3$ .

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Step 10.10.6.4.1

Cancel the common factor.

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \cdot 19^{\frac{3}{3}}}$

Step 10.10.6.4.2

Rewrite the expression.

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \cdot 19^{1}}$

Step 10.10.6.5

Evaluate the exponent.

$\frac{2 \sqrt[3]{511} {\sqrt[3]{19}}^{2}}{3 \cdot 19}$

Step 10.11

Simplify the numerator.

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Step 10.11.1

Rewrite ${\sqrt[3]{19}}^{2}$ as $\sqrt[3]{19^{2}}$ .

$\frac{2 \sqrt[3]{511} \sqrt[3]{19^{2}}}{3 \cdot 19}$

Step 10.11.2

Raise $19$ to the power of $2$ .

$\frac{2 \sqrt[3]{511} \sqrt[3]{361}}{3 \cdot 19}$

Step 10.11.3

Combine exponents.

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Step 10.11.3.1

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{361 \cdot 511}}{3 \cdot 19}$

Step 10.11.3.2

Multiply $361$ by $511$ .

$\frac{2 \sqrt[3]{184471}}{3 \cdot 19}$

Step 10.12

Multiply $3$ by $19$ .

$\frac{2 \sqrt[3]{184471}}{57}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt[3]{184471}}{57}$

Decimal Form:

$1.99739752 \dots$