Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\sqrt[3]{\frac{1 - {(lim_{x \to 3} x)}^{2}}{lim_{x \to 3} 1 - x^{3}}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\sqrt[3]{\frac{1 - {(lim_{x \to 3} x)}^{2}}{lim_{x \to 3} 1 - lim_{x \to 3} x^{3}}}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\sqrt[3]{\frac{1 - {(lim_{x \to 3} x)}^{2}}{1 - lim_{x \to 3} x^{3}}}$

Step 8

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

$\sqrt[3]{\frac{1 - {(lim_{x \to 3} x)}^{2}}{1 - {(lim_{x \to 3} x)}^{3}}}$

Step 9

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\sqrt[3]{\frac{1 - 3^{2}}{1 - {(lim_{x \to 3} x)}^{3}}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 10

Simplify the answer.

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Step 10.1

Raise $3$ to the power of $2$ .

$\sqrt[3]{\frac{1 - 1 \cdot 9}{1 - 3^{3}}}$

Step 10.2

Multiply $- 1$ by $9$ .

$\sqrt[3]{\frac{1 - 9}{1 - 3^{3}}}$

Step 10.3

Subtract $9$ from $1$ .

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

Step 10.4

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

$\sqrt[3]{\frac{- 8}{1 - 1 \cdot 27}}$

Step 10.5

Multiply $- 1$ by $27$ .

$\sqrt[3]{\frac{- 8}{1 - 27}}$

Step 10.6

Subtract $27$ from $1$ .

$\sqrt[3]{\frac{- 8}{- 26}}$

Step 10.7

Cancel the common factor of $- 8$ and $- 26$ .

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Step 10.7.1

Factor $- 2$ out of $- 8$ .

$\sqrt[3]{\frac{- 2 (4)}{- 26}}$

Step 10.7.2

Cancel the common factors.

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Step 10.7.2.1

Factor $- 2$ out of $- 26$ .

$\sqrt[3]{\frac{- 2 \cdot 4}{- 2 \cdot 13}}$

Step 10.7.2.2

Cancel the common factor.

$\sqrt[3]{\frac{- 2 \cdot 4}{- 2 \cdot 13}}$

Step 10.7.2.3

Rewrite the expression.

$\sqrt[3]{\frac{4}{13}}$

Step 10.8

Rewrite $\sqrt[3]{\frac{4}{13}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{13}}$ .

$\frac{\sqrt[3]{4}}{\sqrt[3]{13}}$

Step 10.9

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{13}}$ by $\frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$ .

$\frac{\sqrt[3]{4}}{\sqrt[3]{13}} \cdot \frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$

Step 10.10

Combine and simplify the denominator.

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Step 10.10.1

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{13}}$ by $\frac{{\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{2}}$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{\sqrt[3]{13} {\sqrt[3]{13}}^{2}}$

Step 10.10.2

Raise $\sqrt[3]{13}$ to the power of $1$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{1} {\sqrt[3]{13}}^{2}}$

Step 10.10.3

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

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{1 + 2}}$

Step 10.10.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{{\sqrt[3]{13}}^{3}}$

Step 10.10.5

Rewrite ${\sqrt[3]{13}}^{3}$ as $13$ .

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Step 10.10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{13}$ as $13^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{{(13^{\frac{1}{3}})}^{3}}$

Step 10.10.5.2

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

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{13^{\frac{1}{3} \cdot 3}}$

Step 10.10.5.3

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

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{13^{\frac{3}{3}}}$

Step 10.10.5.4

Cancel the common factor of $3$ .

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Step 10.10.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{13^{\frac{3}{3}}}$

Step 10.10.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{13^{1}}$

Step 10.10.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{4} {\sqrt[3]{13}}^{2}}{13}$

Step 10.11

Simplify the numerator.

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Step 10.11.1

Rewrite ${\sqrt[3]{13}}^{2}$ as $\sqrt[3]{13^{2}}$ .

$\frac{\sqrt[3]{4} \sqrt[3]{13^{2}}}{13}$

Step 10.11.2

Raise $13$ to the power of $2$ .

$\frac{\sqrt[3]{4} \sqrt[3]{169}}{13}$

Step 10.12

Simplify the numerator.

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Step 10.12.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{4 \cdot 169}}{13}$

Step 10.12.2

Multiply $4$ by $169$ .

$\frac{\sqrt[3]{676}}{13}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{676}}{13}$

Decimal Form:

$0.67510638 \dots$