Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Add $1$ and $1$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $1$ as $1^{3}$ .

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

Step 3.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = x$ .

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

Step 3.2.3

Simplify.

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Step 3.2.3.1

One to any power is one.

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

Step 3.2.3.2

Multiply $x$ by $1$ .

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

Step 3.3

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{(1 - x) (1 + x + x^{2})}}$ by $\frac{{\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{2}}{{\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{2}}$ .

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{(1 - x) (1 + x + x^{2})}}$ by $\frac{{\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{2}}{{\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{2}}$ .

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

Step 3.4.2

Raise $\sqrt[3]{(1 - x) (1 + x + x^{2})}$ to the power of $1$ .

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

Step 3.4.3

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

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

Step 3.4.4

Add $1$ and $2$ .

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

Step 3.4.5

Rewrite ${\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{3}$ as $(1 - x) (1 + x + x^{2})$ .

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Step 3.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{(1 - x) (1 + x + x^{2})}$ as ${((1 - x) (1 + x + x^{2}))}^{\frac{1}{3}}$ .

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

Step 3.4.5.2

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

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

Step 3.4.5.3

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

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

Step 3.4.5.4

Cancel the common factor of $3$ .

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Step 3.4.5.4.1

Cancel the common factor.

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

Step 3.4.5.4.2

Rewrite the expression.

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

Step 3.4.5.5

Simplify.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Rewrite ${\sqrt[3]{(1 - x) (1 + x + x^{2})}}^{2}$ as $\sqrt[3]{{((1 - x) (1 + x + x^{2}))}^{2}}$ .

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

Step 3.5.2

Apply the product rule to $(1 - x) (1 + x + x^{2})$ .

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

Step 3.6

Combine using the product rule for radicals.

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