Calculus Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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Step 2.2.1

Subtract $x^{2} \sqrt[3]{x^{2} (x - 8)}$ from $x^{2} \sqrt[3]{x^{2} (x - 8)}$ .

$lim_{x \to \infty} \frac{x^{3} - 8 x^{2} + 0 - x^{3}}{{\sqrt[3]{x^{3} - 8 x^{2}}}^{2} - \sqrt[3]{x^{3} - 8 x^{2}} (- x) + {(- x)}^{2}}$

Step 2.2.2

Add $x^{3} - 8 x^{2}$ and $0$ .

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

Step 2.2.3

Subtract $x^{3}$ from $x^{3}$ .

$lim_{x \to \infty} \frac{- 8 x^{2} + 0}{{\sqrt[3]{x^{3} - 8 x^{2}}}^{2} - \sqrt[3]{x^{3} - 8 x^{2}} (- x) + {(- x)}^{2}}$

Step 2.2.4

Add $- 8 x^{2}$ and $0$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Factor $x^{2}$ out of $x^{3} - 8 x^{2}$ .

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Step 3.1.1.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 3.1.1.2

Factor $x^{2}$ out of $- 8 x^{2}$ .

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

Step 3.1.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 8$ .

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

Step 3.1.2

Rewrite ${\sqrt[3]{x^{2} (x - 8)}}^{2}$ as $\sqrt[3]{{(x^{2} (x - 8))}^{2}}$ .

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

Step 3.1.3

Apply the product rule to $x^{2} (x - 8)$ .

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

Step 3.1.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.1.4.1

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

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

Step 3.1.4.2

Multiply $2$ by $2$ .

$lim_{x \to \infty} \frac{- 8 x^{2}}{\sqrt[3]{x^{4} {(x - 8)}^{2}} - \sqrt[3]{x^{3} - 8 x^{2}} (- x) + {(- x)}^{2}}$

Step 3.1.5

Rewrite $x^{4} {(x - 8)}^{2}$ as $x^{3} (x {(x + {(- 2)}^{3})}^{2})$ .

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Step 3.1.5.1

Factor out $x^{3}$ .

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

Step 3.1.5.2

Rewrite $- 8$ as ${(- 2)}^{3}$ .

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

Step 3.1.5.3

Add parentheses.

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

Step 3.1.6

Pull terms out from under the radical.

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

Step 3.1.7

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

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

Step 3.1.8

Factor $x^{2}$ out of $x^{3} - 8 x^{2}$ .

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Step 3.1.8.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 3.1.8.2

Factor $x^{2}$ out of $- 8 x^{2}$ .

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

Step 3.1.8.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 8$ .

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

Step 3.1.9

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

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Step 3.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.9.2

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

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

Step 3.1.10

Apply the product rule to $- x$ .

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

Step 3.1.11

Raise $- 1$ to the power of $2$ .

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

Step 3.1.12

Multiply $x^{2}$ by $1$ .

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

Step 3.2

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

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

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x^{2}$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Simplify each term.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Cancel the common factor of $x$ .

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Step 5.3.1.1

Cancel the common factor.

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

Step 5.3.1.2

Rewrite the expression.

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

Step 5.3.2

Move the negative in front of the fraction.

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

Step 5.4

Rewrite ${(1 - \frac{8}{x})}^{2}$ as $(1 - \frac{8}{x}) (1 - \frac{8}{x})$ .

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

Step 6

Expand $(1 - \frac{8}{x}) (1 - \frac{8}{x})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $1$ by $1$ .

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

Step 7.1.2

Multiply $- \frac{8}{x}$ by $1$ .

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

Step 7.1.3

Multiply $- 1$ by $1$ .

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

Step 7.1.4

Multiply $- \frac{8}{x} (- \frac{8}{x})$ .

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Step 7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.4.2

Multiply $\frac{8}{x}$ by $1$ .

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

Step 7.1.4.3

Multiply $\frac{8}{x}$ by $\frac{8}{x}$ .

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

Step 7.1.4.4

Multiply $8$ by $8$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{8}{x} - \frac{8}{x} + \frac{64}{x \cdot x})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 7.1.4.5

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

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{8}{x} - \frac{8}{x} + \frac{64}{x^{1} x})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 7.1.4.6

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

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{8}{x} - \frac{8}{x} + \frac{64}{x^{1} x^{1}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 7.1.4.7

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

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{8}{x} - \frac{8}{x} + \frac{64}{x^{1 + 1}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 7.1.4.8

Add $1$ and $1$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{8}{x} - \frac{8}{x} + \frac{64}{x^{2}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 7.2

Subtract $\frac{8}{x}$ from $- \frac{8}{x}$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - 2 \frac{8}{x} + \frac{64}{x^{2}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 8

Simplify each term.

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Step 8.1

Multiply $- 2 \frac{8}{x}$ .

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Step 8.1.1

Combine $- 2$ and $\frac{8}{x}$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 + \frac{- 2 \cdot 8}{x} + \frac{64}{x^{2}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 8.1.2

Multiply $- 2$ by $8$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 + \frac{- 16}{x} + \frac{64}{x^{2}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 8.2

Move the negative in front of the fraction.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x (1 - \frac{16}{x} + \frac{64}{x^{2}})} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 9

Apply the distributive property.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x \cdot 1 + x \cdot - 1 \frac{16}{x} + x \frac{64}{x^{2}}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 10

Simplify.

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Step 10.1

Multiply $x$ by $1$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x + x \cdot - 1 \frac{16}{x} + x \frac{64}{x^{2}}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 10.2

Rewrite using the commutative property of multiplication.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - x \frac{16}{x} + x \frac{64}{x^{2}}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 10.3

Cancel the common factor of $x$ .

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Step 10.3.1

Factor $x$ out of $x^{2}$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - x \frac{16}{x} + x \frac{64}{x \cdot x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 10.3.2

Cancel the common factor.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - x \frac{16}{x} + x \frac{64}{x \cdot x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 10.3.3

Rewrite the expression.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - x \frac{16}{x} + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 11

Simplify each term.

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Step 11.1

Cancel the common factor of $x$ .

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Step 11.1.1

Factor $x$ out of $- x$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x + x \cdot - 1 \frac{16}{x} + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 11.1.2

Cancel the common factor.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x + x \cdot - 1 \frac{16}{x} + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 11.1.3

Rewrite the expression.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 1 \cdot 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 11.2

Multiply $- 1$ by $16$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} (x - 8)}}{x} + 1}$

Step 12

Apply the distributive property.

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} x + x^{2} \cdot - 8}}{x} + 1}$

Step 13

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 13.1

Multiply $x^{2}$ by $x$ .

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Step 13.1.1

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

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{2} x^{1} + x^{2} \cdot - 8}}{x} + 1}$

Step 13.1.2

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

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{2 + 1} + x^{2} \cdot - 8}}{x} + 1}$

Step 13.2

Add $2$ and $1$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{3} + x^{2} \cdot - 8}}{x} + 1}$

Step 14

Move $- 8$ to the left of $x^{2}$ .

$- 8 lim_{x \to \infty} \frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{3} - 8 x^{2}}}{x} + 1}$

Step 15

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x \sqrt[3]{x - 16 + \frac{64}{x}} + \frac{\sqrt[3]{x^{3} - 8 x^{2}}}{x} + 1}$ approaches $0$ .

$- 8 \cdot 0$

Step 16

Multiply $- 8$ by $0$ .

$0$