Calculus Examples

$lim_{x \to \infty} \sqrt{81 x^{2} + x} - 9 x$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to \infty} \frac{(\sqrt{81 x^{2} + x} - 9 x) (\sqrt{81 x^{2} + x} + 9 x)}{\sqrt{81 x^{2} + x} + 9 x}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty} \frac{{\sqrt{81 x^{2} + x}}^{2} + \sqrt{81 x^{2} + x} (9 x) + \sqrt{81 x^{2} + x} (- 9 x) - 81 x^{2}}{\sqrt{81 x^{2} + x} + 9 x}$

Step 2.2

Simplify.

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Step 2.2.1

Subtract $81 x^{2}$ from $81 x^{2}$ .

$lim_{x \to \infty} \frac{x + 0}{\sqrt{81 x^{2} + x} + 9 x}$

Step 2.2.2

Add $x$ and $0$ .

$lim_{x \to \infty} \frac{x}{\sqrt{81 x^{2} + x} + 9 x}$

Step 3

Factor $x$ out of $81 x^{2} + x$ .

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Step 3.1

Factor $x$ out of $81 x^{2}$ .

$lim_{x \to \infty} \frac{x}{\sqrt{x (81 x) + x} + 9 x}$

Step 3.2

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

$lim_{x \to \infty} \frac{x}{\sqrt{x (81 x) + x^{1}} + 9 x}$

Step 3.3

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{x}{\sqrt{x (81 x) + x \cdot 1} + 9 x}$

Step 3.4

Factor $x$ out of $x (81 x) + x \cdot 1$ .

$lim_{x \to \infty} \frac{x}{\sqrt{x (81 x + 1)} + 9 x}$

Step 4

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

$lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt{\frac{x (81 x + 1)}{x^{2}}} + \frac{9 x}{x}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt{\frac{x (81 x + 1)}{x^{2}}} + \frac{9 x}{x}}$

Step 5.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{x (81 x + 1)}{x^{2}}} + \frac{9 x}{x}}$

Step 5.2

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{x (81 x + 1)}{x^{2}}} + 9}$

Step 6

Cancel the common factors.

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Step 6.1

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

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{x (81 x + 1)}{x \cdot x}} + 9}$

Step 6.2

Cancel the common factor.

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{x (81 x + 1)}{x \cdot x}} + 9}$

Step 6.3

Rewrite the expression.

$lim_{x \to \infty} \frac{1}{\sqrt{\frac{81 x + 1}{x}} + 9}$

Step 7

Evaluate the limit.

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Step 7.1

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

$\frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \sqrt{\frac{81 x + 1}{x}} + 9}$

Step 7.2

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

$\frac{1}{lim_{x \to \infty} \sqrt{\frac{81 x + 1}{x}} + 9}$

Step 7.3

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

$\frac{1}{lim_{x \to \infty} \sqrt{\frac{81 x + 1}{x}} + lim_{x \to \infty} 9}$

Step 7.4

Move the limit under the radical sign.

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{81 x + 1}{x}} + lim_{x \to \infty} 9}$

Step 8

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

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{\frac{81 x}{x} + \frac{1}{x}}{\frac{x}{x}}} + lim_{x \to \infty} 9}$

Step 9

Evaluate the limit.

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Step 9.1

Cancel the common factor of $x$ .

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Step 9.1.1

Cancel the common factor.

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{\frac{81 x}{x} + \frac{1}{x}}{\frac{x}{x}}} + lim_{x \to \infty} 9}$

Step 9.1.2

Divide $81$ by $1$ .

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{81 + \frac{1}{x}}{\frac{x}{x}}} + lim_{x \to \infty} 9}$

Step 9.2

Cancel the common factor of $x$ .

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Step 9.2.1

Cancel the common factor.

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{81 + \frac{1}{x}}{\frac{x}{x}}} + lim_{x \to \infty} 9}$

Step 9.2.2

Rewrite the expression.

$\frac{1}{\sqrt{lim_{x \to \infty} \frac{81 + \frac{1}{x}}{1}} + lim_{x \to \infty} 9}$

Step 9.3

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

$\frac{1}{\sqrt{\frac{lim_{x \to \infty} 81 + \frac{1}{x}}{lim_{x \to \infty} 1}} + lim_{x \to \infty} 9}$

Step 9.4

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

$\frac{1}{\sqrt{\frac{lim_{x \to \infty} 81 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}} + lim_{x \to \infty} 9}$

Step 9.5

Evaluate the limit of $81$ which is constant as $x$ approaches $\infty$ .

$\frac{1}{\sqrt{\frac{81 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}} + lim_{x \to \infty} 9}$

Step 10

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

$\frac{1}{\sqrt{\frac{81 + 0}{lim_{x \to \infty} 1}} + lim_{x \to \infty} 9}$

Step 11

Evaluate the limit.

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Step 11.1

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

$\frac{1}{\sqrt{\frac{81 + 0}{1}} + lim_{x \to \infty} 9}$

Step 11.2

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

$\frac{1}{\sqrt{\frac{81 + 0}{1}} + 9}$

Step 11.3

Simplify the answer.

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Step 11.3.1

Divide $81 + 0$ by $1$ .

$\frac{1}{\sqrt{81 + 0} + 9}$

Step 11.3.2

Simplify the denominator.

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Step 11.3.2.1

Add $81$ and $0$ .

$\frac{1}{\sqrt{81} + 9}$

Step 11.3.2.2

Rewrite $81$ as $9^{2}$ .

$\frac{1}{\sqrt{9^{2}} + 9}$

Step 11.3.2.3

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

$\frac{1}{9 + 9}$

Step 11.3.2.4

Add $9$ and $9$ .

$\frac{1}{18}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{18}$

Decimal Form:

$0.0\overline{5}$