Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{9 x^{6} - x}}{x^{3} + 5}$

Step 1

Factor $x$ out of $9 x^{6} - x$ .

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Step 1.1

Factor $x$ out of $9 x^{6}$ .

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

Step 1.2

Factor $x$ out of $- x$ .

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

Step 1.3

Factor $x$ out of $x (9 x^{5}) + x \cdot - 1$ .

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $\sqrt{x^{6}} = x^{3}$ .

$lim_{x \to \infty} \frac{\sqrt{\frac{x (9 x^{5} - 1)}{x^{6}}}}{\frac{x^{3}}{x^{3}} + \frac{5}{x^{3}}}$

Step 3

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

$lim_{x \to \infty} \frac{\sqrt{\frac{x (9 x^{5} - 1)}{x^{6}}}}{1 + \frac{5}{x^{3}}}$

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\sqrt{\frac{9 x^{5} - 1}{x^{5}}}}{1 + \frac{5}{x^{3}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{lim_{x \to \infty} \sqrt{\frac{9 x^{5} - 1}{x^{5}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 5.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to \infty} \frac{9 x^{5} - 1}{x^{5}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 6

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{5}$ .

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{9 x^{5}}{x^{5}} - \frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7

Evaluate the limit.

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Step 7.1

Cancel the common factor of $x^{5}$ .

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Step 7.1.1

Cancel the common factor.

$\frac{\sqrt{lim_{x \to \infty} \frac{\frac{9 x^{5}}{x^{5}} - \frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.1.2

Divide $9$ by $1$ .

$\frac{\sqrt{lim_{x \to \infty} \frac{9 - \frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.2

Cancel the common factor of $x^{5}$ .

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Step 7.2.1

Cancel the common factor.

$\frac{\sqrt{lim_{x \to \infty} \frac{9 - \frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}}}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.2.2

Rewrite the expression.

$\frac{\sqrt{lim_{x \to \infty} \frac{9 - \frac{1}{x^{5}}}{1}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.3

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

$\frac{\sqrt{\frac{lim_{x \to \infty} 9 - \frac{1}{x^{5}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.4

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

$\frac{\sqrt{\frac{lim_{x \to \infty} 9 - lim_{x \to \infty} \frac{1}{x^{5}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 7.5

Evaluate the limit of $9$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{\frac{9 - lim_{x \to \infty} \frac{1}{x^{5}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1 + \frac{5}{x^{3}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{5}}$ approaches $0$ .

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

Step 9

Evaluate the limit.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 10

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

$\frac{\sqrt{\frac{9 - 0}{1}}}{1 + 5 \cdot 0}$

Step 11

Simplify the answer.

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Step 11.1

Divide $9 - 0$ by $1$ .

$\frac{\sqrt{9 - 0}}{1 + 5 \cdot 0}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Multiply $- 1$ by $0$ .

$\frac{\sqrt{9 + 0}}{1 + 5 \cdot 0}$

Step 11.2.2

Add $9$ and $0$ .

$\frac{\sqrt{9}}{1 + 5 \cdot 0}$

Step 11.2.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}}}{1 + 5 \cdot 0}$

Step 11.2.4

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

$\frac{3}{1 + 5 \cdot 0}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

Multiply $5$ by $0$ .

$\frac{3}{1 + 0}$

Step 11.3.2

Add $1$ and $0$ .

$\frac{3}{1}$

Step 11.4

Divide $3$ by $1$ .

$3$