Calculus Examples

$lim_{x \to \infty} \sqrt{x^{6} - 5 x^{3}} - x^{3}$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to \infty} \frac{(\sqrt{x^{6} - 5 x^{3}} - x^{3}) (\sqrt{x^{6} - 5 x^{3}} + x^{3})}{\sqrt{x^{6} - 5 x^{3}} + x^{3}}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty} \frac{{\sqrt{x^{6} - 5 x^{3}}}^{2} + \sqrt{x^{6} - 5 x^{3}} x^{3} + \sqrt{x^{6} - 5 x^{3}} (- x^{3}) - x^{6}}{\sqrt{x^{6} - 5 x^{3}} + x^{3}}$

Step 2.2

Simplify.

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Step 2.2.1

Subtract $x^{6}$ from $x^{6}$ .

$lim_{x \to \infty} \frac{- 5 x^{3} + 0}{\sqrt{x^{6} - 5 x^{3}} + x^{3}}$

Step 2.2.2

Add $- 5 x^{3}$ and $0$ .

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{6} - 5 x^{3}} + x^{3}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Factor $x^{3}$ out of $x^{6} - 5 x^{3}$ .

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Step 3.1.1.1

Factor $x^{3}$ out of $x^{6}$ .

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{3} x^{3} - 5 x^{3}} + x^{3}}$

Step 3.1.1.2

Factor $x^{3}$ out of $- 5 x^{3}$ .

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{3} x^{3} + x^{3} \cdot - 5} + x^{3}}$

Step 3.1.1.3

Factor $x^{3}$ out of $x^{3} x^{3} + x^{3} \cdot - 5$ .

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{3} (x^{3} - 5)} + x^{3}}$

Step 3.1.2

Rewrite $x^{3} (x^{3} - 5)$ as $x^{2} (x (x^{3} - 5))$ .

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Step 3.1.2.1

Factor out $x^{2}$ .

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{2} x (x^{3} - 5)} + x^{3}}$

Step 3.1.2.2

Add parentheses.

$lim_{x \to \infty} \frac{- 5 x^{3}}{\sqrt{x^{2} (x (x^{3} - 5))} + x^{3}}$

Step 3.1.3

Pull terms out from under the radical.

$lim_{x \to \infty} \frac{- 5 x^{3}}{x \sqrt{x (x^{3} - 5)} + x^{3}}$

Step 3.2

Move the term $- 5$ outside of the limit because it is constant with respect to $x$ .

$- 5 lim_{x \to \infty} \frac{x^{3}}{x \sqrt{x (x^{3} - 5)} + x^{3}}$

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$- 5 lim_{x \to \infty} \frac{\frac{x^{3}}{x^{3}}}{\frac{x \sqrt{x (x^{3} - 5)}}{x^{3}} + \frac{x^{3}}{x^{3}}}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x^{3}$ .

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Step 5.1.1

Cancel the common factor.

$- 5 lim_{x \to \infty} \frac{\frac{x^{3}}{x^{3}}}{\frac{x \sqrt{x (x^{3} - 5)}}{x^{3}} + \frac{x^{3}}{x^{3}}}$

Step 5.1.2

Rewrite the expression.

$- 5 lim_{x \to \infty} \frac{1}{\frac{x \sqrt{x (x^{3} - 5)}}{x^{3}} + \frac{x^{3}}{x^{3}}}$

Step 5.2

Simplify each term.

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x (x^{3} - 5)}}{x^{2}} + 1}$

Step 5.3

Apply the distributive property.

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x \cdot x^{3} + x \cdot - 5}}{x^{2}} + 1}$

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x^{1} x^{3} + x \cdot - 5}}{x^{2}} + 1}$

Step 6.1.2

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

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x^{1 + 3} + x \cdot - 5}}{x^{2}} + 1}$

Step 6.2

Add $1$ and $3$ .

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x^{4} + x \cdot - 5}}{x^{2}} + 1}$

Step 7

Evaluate the limit.

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Step 7.1

Move $- 5$ to the left of $x$ .

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x^{4} - 5 x}}{x^{2}} + 1}$

Step 7.2

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

$- 5 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{\sqrt{x^{4} - 5 x}}{x^{2}} + 1}$

Step 7.3

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x^{4} - 5 x}}{x^{2}} + 1}$

Step 7.4

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x^{4} - 5 x}}{x^{2}} + lim_{x \to \infty} 1}$

Step 7.5

Factor $x$ out of $x^{4} - 5 x$ .

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Step 7.5.1

Factor $x$ out of $x^{4}$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x \cdot x^{3} - 5 x}}{x^{2}} + lim_{x \to \infty} 1}$

Step 7.5.2

Factor $x$ out of $- 5 x$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x \cdot x^{3} + x \cdot - 5}}{x^{2}} + lim_{x \to \infty} 1}$

Step 7.5.3

Factor $x$ out of $x \cdot x^{3} + x \cdot - 5$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x (x^{3} - 5)}}{x^{2}} + lim_{x \to \infty} 1}$

Step 8

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $\sqrt{x^{4}} = x^{2}$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x (x^{3} - 5)}{x^{4}}}}{\frac{x^{2}}{x^{2}}} + lim_{x \to \infty} 1}$

Step 9

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x (x^{3} - 5)}{x^{4}}}}{1} + lim_{x \to \infty} 1}$

Step 10

Cancel the common factors.

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Step 10.1

Factor $x$ out of $x^{4}$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x (x^{3} - 5)}{x \cdot x^{3}}}}{1} + lim_{x \to \infty} 1}$

Step 10.2

Cancel the common factor.

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x (x^{3} - 5)}{x \cdot x^{3}}}}{1} + lim_{x \to \infty} 1}$

Step 10.3

Rewrite the expression.

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x^{3} - 5}{x^{3}}}}{1} + lim_{x \to \infty} 1}$

Step 11

Evaluate the limit.

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Step 11.1

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

$- 5 \frac{1}{\frac{lim_{x \to \infty} \sqrt{\frac{x^{3} - 5}{x^{3}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 11.2

Move the limit under the radical sign.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{x^{3} - 5}{x^{3}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 12

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{\frac{x^{3}}{x^{3}} + \frac{- 5}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13

Evaluate the limit.

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Step 13.1

Simplify each term.

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Step 13.1.1

Cancel the common factor of $x^{3}$ .

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Step 13.1.1.1

Cancel the common factor.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{\frac{x^{3}}{x^{3}} + \frac{- 5}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.1.1.2

Rewrite the expression.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{1 + \frac{- 5}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.1.2

Move the negative in front of the fraction.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{1 - \frac{5}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.2

Cancel the common factor of $x^{3}$ .

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Step 13.2.1

Cancel the common factor.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{1 - \frac{5}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.2.2

Rewrite the expression.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} \frac{1 - \frac{5}{x^{3}}}{1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.3

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

$- 5 \frac{1}{\frac{\sqrt{\frac{lim_{x \to \infty} 1 - \frac{5}{x^{3}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.4

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

$- 5 \frac{1}{\frac{\sqrt{\frac{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.5

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

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 13.6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - 5 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 14

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - 5 \cdot 0}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 15

Evaluate the limit.

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Step 15.1

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

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - 5 \cdot 0}{1}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 15.2

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

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - 5 \cdot 0}{1}}}{1} + lim_{x \to \infty} 1}$

Step 15.3

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

$- 5 \frac{1}{\frac{\sqrt{\frac{1 - 5 \cdot 0}{1}}}{1} + 1}$

Step 15.4

Simplify the answer.

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Step 15.4.1

Divide $1 - 5 \cdot 0$ by $1$ .

$- 5 \frac{1}{\frac{\sqrt{1 - 5 \cdot 0}}{1} + 1}$

Step 15.4.2

Divide $\sqrt{1 - 5 \cdot 0}$ by $1$ .

$- 5 \frac{1}{\sqrt{1 - 5 \cdot 0} + 1}$

Step 15.4.3

Simplify the denominator.

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Step 15.4.3.1

Multiply $- 5$ by $0$ .

$- 5 \frac{1}{\sqrt{1 + 0} + 1}$

Step 15.4.3.2

Add $1$ and $0$ .

$- 5 \frac{1}{\sqrt{1} + 1}$

Step 15.4.3.3

Any root of $1$ is $1$ .

$- 5 \frac{1}{1 + 1}$

Step 15.4.3.4

Add $1$ and $1$ .

$- 5 (\frac{1}{2})$

Step 15.4.4

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{- 5}{2}$

Step 15.4.5

Move the negative in front of the fraction.

$- \frac{5}{2}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{2}$

Decimal Form:

$- 2.5$