Calculus Examples

$\frac{lim_{x \to 8} x}{\sqrt[3]{x^{2} - 1}}$

Step 1

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

$\frac{8}{\sqrt[3]{x^{2} - 1}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{8}{\sqrt[3]{x^{2} - 1^{2}}}$

Step 2.1.2

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

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

Step 2.2

Multiply $\frac{8}{\sqrt[3]{(x + 1) (x - 1)}}$ by $\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$ .

$\frac{8}{\sqrt[3]{(x + 1) (x - 1)}} \cdot \frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{8}{\sqrt[3]{(x + 1) (x - 1)}}$ by $\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$ .

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{\sqrt[3]{(x + 1) (x - 1)} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

Step 2.3.2

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

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{1} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

Step 2.3.3

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

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{1 + 2}}$

Step 2.3.4

Add $1$ and $2$ .

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{3}}$

Step 2.3.5

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

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Step 2.3.5.1

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

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{({((x + 1) (x - 1))}^{\frac{1}{3}})}^{3}}$

Step 2.3.5.2

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

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{1}{3} \cdot 3}}$

Step 2.3.5.3

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

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{3}{3}}}$

Step 2.3.5.4

Cancel the common factor of $3$ .

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Step 2.3.5.4.1

Cancel the common factor.

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{3}{3}}}$

Step 2.3.5.4.2

Rewrite the expression.

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{1}}$

Step 2.3.5.5

Simplify.

$\frac{8 {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{(x + 1) (x - 1)}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

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

$\frac{8 \sqrt[3]{{((x + 1) (x - 1))}^{2}}}{(x + 1) (x - 1)}$

Step 2.4.2

Apply the product rule to $(x + 1) (x - 1)$ .

$\frac{8 \sqrt[3]{{(x + 1)}^{2} {(x - 1)}^{2}}}{(x + 1) (x - 1)}$