Calculus Examples

$lim_{x \to \infty} \sqrt{\frac{3 x + 2}{6 x - 9}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to \infty} \frac{3 x + 2}{6 x - 9}}$

Step 2

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

$\sqrt{lim_{x \to \infty} \frac{\frac{3 x}{x} + \frac{2}{x}}{\frac{6 x}{x} + \frac{- 9}{x}}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

$\sqrt{lim_{x \to \infty} \frac{\frac{3 x}{x} + \frac{2}{x}}{\frac{6 x}{x} + \frac{- 9}{x}}}$

Step 3.1.2

Divide $3$ by $1$ .

$\sqrt{lim_{x \to \infty} \frac{3 + \frac{2}{x}}{\frac{6 x}{x} + \frac{- 9}{x}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x$ .

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Step 3.2.1.1

Cancel the common factor.

$\sqrt{lim_{x \to \infty} \frac{3 + \frac{2}{x}}{\frac{6 x}{x} + \frac{- 9}{x}}}$

Step 3.2.1.2

Divide $6$ by $1$ .

$\sqrt{lim_{x \to \infty} \frac{3 + \frac{2}{x}}{6 + \frac{- 9}{x}}}$

Step 3.2.2

Move the negative in front of the fraction.

$\sqrt{lim_{x \to \infty} \frac{3 + \frac{2}{x}}{6 - \frac{9}{x}}}$

Step 3.3

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

$\sqrt{\frac{lim_{x \to \infty} 3 + \frac{2}{x}}{lim_{x \to \infty} 6 - \frac{9}{x}}}$

Step 3.4

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

$\sqrt{\frac{lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{2}{x}}{lim_{x \to \infty} 6 - \frac{9}{x}}}$

Step 3.5

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$\sqrt{\frac{3 + lim_{x \to \infty} \frac{2}{x}}{lim_{x \to \infty} 6 - \frac{9}{x}}}$

Step 3.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{3 + 2 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 6 - \frac{9}{x}}}$

Step 4

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

$\sqrt{\frac{3 + 2 \cdot 0}{lim_{x \to \infty} 6 - \frac{9}{x}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\sqrt{\frac{3 + 2 \cdot 0}{lim_{x \to \infty} 6 - lim_{x \to \infty} \frac{9}{x}}}$

Step 5.2

Evaluate the limit of $6$ which is constant as $x$ approaches $\infty$ .

$\sqrt{\frac{3 + 2 \cdot 0}{6 - lim_{x \to \infty} \frac{9}{x}}}$

Step 5.3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{3 + 2 \cdot 0}{6 - 9 lim_{x \to \infty} \frac{1}{x}}}$

Step 6

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

$\sqrt{\frac{3 + 2 \cdot 0}{6 - 9 \cdot 0}}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $2$ by $0$ .

$\sqrt{\frac{3 + 0}{6 - 9 \cdot 0}}$

Step 7.2

Add $3$ and $0$ .

$\sqrt{\frac{3}{6 - 9 \cdot 0}}$

Step 7.3

Multiply $- 9$ by $0$ .

$\sqrt{\frac{3}{6 + 0}}$

Step 7.4

Add $6$ and $0$ .

$\sqrt{\frac{3}{6}}$

Step 7.5

Cancel the common factor of $3$ and $6$ .

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Step 7.5.1

Factor $3$ out of $3$ .

$\sqrt{\frac{3 (1)}{6}}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $3$ out of $6$ .

$\sqrt{\frac{3 \cdot 1}{3 \cdot 2}}$

Step 7.5.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 1}{3 \cdot 2}}$

Step 7.5.2.3

Rewrite the expression.

$\sqrt{\frac{1}{2}}$

Step 7.6

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{1}}{\sqrt{2}}$

Step 7.7

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}}$

Step 7.8

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.9

Combine and simplify the denominator.

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Step 7.9.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.9.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.9.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.9.4

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

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

Step 7.9.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 7.9.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 7.9.6.2

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

$\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 7.9.6.3

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

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.9.6.4

Cancel the common factor of $2$ .

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Step 7.9.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.9.6.4.2

Rewrite the expression.

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

Step 7.9.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$