Calculus Examples

lim x \to \infty \frac{x - 5}{\sqrt[3]{27 x^{3} + 12}}

$lim_{x \to \infty} \frac{x - 5}{\sqrt[3]{27 x^{3} + 12}}$

Step 1

Factor

3

$3$ out of

27 x^{3} + 12

$27 x^{3} + 12$ .

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Step 1.1

Factor

3

$3$ out of

27 x^{3}

$27 x^{3}$ .

lim x \to \infty \frac{x - 5}{\sqrt[3]{3 (9 x^{3}) + 12}}

$lim_{x \to \infty} \frac{x - 5}{\sqrt[3]{3 (9 x^{3}) + 12}}$

Step 1.2

Factor

3

$3$ out of

12

$12$ .

lim x \to \infty \frac{x - 5}{\sqrt[3]{3 (9 x^{3}) + 3 (4)}}

$lim_{x \to \infty} \frac{x - 5}{\sqrt[3]{3 (9 x^{3}) + 3 (4)}}$

Step 1.3

Factor

3

$3$ out of

3 (9 x^{3}) + 3 (4)

$3 (9 x^{3}) + 3 (4)$ .

lim x \to \infty \frac{x - 5}{\sqrt[3]{3 (9 x^{3} + 4)}}

$lim_{x \to \infty} \frac{x - 5}{\sqrt[3]{3 (9 x^{3} + 4)}}$

lim x \to \infty \frac{x - 5}{\sqrt[3]{3 (9 x^{3} + 4)}}

$lim_{x \to \infty} \frac{x - 5}{\sqrt[3]{3 (9 x^{3} + 4)}}$

Step 2

Divide the numerator and denominator by the highest power of

x

$x$ in the denominator, which is

x = \sqrt[3]{x^{3}}

$x = \sqrt[3]{x^{3}}$ .

lim x \to \infty \frac{\frac{x}{x} + \frac{- 5}{x}}{\sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$lim_{x \to \infty} \frac{\frac{x}{x} + \frac{- 5}{x}}{\sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

lim x \to \infty \frac{1 - \frac{5}{x}}{\sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$lim_{x \to \infty} \frac{1 - \frac{5}{x}}{\sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as

x

$x$ approaches

\infty

$\infty$ .

\frac{lim x \to \infty 1 - \frac{5}{x}}{lim x \to \infty \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$\frac{lim_{x \to \infty} 1 - \frac{5}{x}}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

\infty

$\infty$ .

\frac{lim x \to \infty 1 - lim x \to \infty \frac{5}{x}}{lim x \to \infty \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$\frac{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{5}{x}}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 3.4

Evaluate the limit of

1

$1$ which is constant as

x

$x$ approaches

\infty

$\infty$ .

\frac{1 - lim x \to \infty \frac{5}{x}}{lim x \to \infty \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$\frac{1 - lim_{x \to \infty} \frac{5}{x}}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 3.5

Move the term

5

$5$ outside of the limit because it is constant with respect to

x

$x$ .

\frac{1 - 5 lim x \to \infty \frac{1}{x}}{lim x \to \infty \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$\frac{1 - 5 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

\frac{1 - 5 lim x \to \infty \frac{1}{x}}{lim x \to \infty \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}

$\frac{1 - 5 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{x}$ approaches

$0$ .

$\frac{1 - 5 \cdot 0}{lim_{x \to \infty} \sqrt[3]{\frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the limit under the radical sign.

$\frac{1 - 5 \cdot 0}{\sqrt[3]{lim_{x \to \infty} \frac{3 (9 x^{3} + 4)}{x^{3}}}}$

Step 5.2

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{9 x^{3} + 4}{x^{3}}}}$

Step 6

Divide the numerator and denominator by the highest power of

$x$ in the denominator, which is

$x^{3}$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{\frac{9 x^{3}}{x^{3}} + \frac{4}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}$

Step 7

Evaluate the limit.

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Step 7.1

Cancel the common factor of

$x^{3}$ .

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Step 7.1.1

Cancel the common factor.

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{\frac{9 x^{3}}{x^{3}} + \frac{4}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}$

Step 7.1.2

Divide

$9$ by

$1$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{9 + \frac{4}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}$

Step 7.2

Cancel the common factor of

$x^{3}$ .

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Step 7.2.1

Cancel the common factor.

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{9 + \frac{4}{x^{3}}}{\frac{x^{3}}{x^{3}}}}}$

Step 7.2.2

Rewrite the expression.

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 lim_{x \to \infty} \frac{9 + \frac{4}{x^{3}}}{1}}}$

Step 7.3

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$\infty$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{lim_{x \to \infty} 9 + \frac{4}{x^{3}}}{lim_{x \to \infty} 1}}}$

Step 7.4

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$\infty$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{lim_{x \to \infty} 9 + lim_{x \to \infty} \frac{4}{x^{3}}}{lim_{x \to \infty} 1}}}$

Step 7.5

Evaluate the limit of

$9$ which is constant as

$x$ approaches

$\infty$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{9 + lim_{x \to \infty} \frac{4}{x^{3}}}{lim_{x \to \infty} 1}}}$

Step 7.6

Move the term

$4$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{9 + 4 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 1}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{x^{3}}$ approaches

$0$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{9 + 4 \cdot 0}{lim_{x \to \infty} 1}}}$

Step 9

Evaluate the limit.

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Step 9.1

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$\infty$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 \frac{9 + 4 \cdot 0}{1}}}$

Step 9.2

Simplify the answer.

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Step 9.2.1

Divide

$9 + 4 \cdot 0$ by

$1$ .

$\frac{1 - 5 \cdot 0}{\sqrt[3]{3 (9 + 4 \cdot 0)}}$

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Multiply

$- 5$ by

$0$ .

$\frac{1 + 0}{\sqrt[3]{3 (9 + 4 \cdot 0)}}$

Step 9.2.2.2

Add

$1$ and

$0$ .

$\frac{1}{\sqrt[3]{3 (9 + 4 \cdot 0)}}$

Step 9.2.3

Simplify the denominator.

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Step 9.2.3.1

Multiply

$4$ by

$0$ .

$\frac{1}{\sqrt[3]{3 (9 + 0)}}$

Step 9.2.3.2

Add

$9$ and

$0$ .

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

Step 9.2.3.3

Multiply

$3$ by

$9$ .

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

Step 9.2.3.4

Rewrite

$27$ as

$3^{3}$ .

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

Step 9.2.3.5

Pull terms out from under the radical, assuming real numbers.

$\frac{1}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$