Calculus Examples

$lim_{x \to - \infty} \sqrt{\frac{4 x^{2} + 2 x - 1}{9 x^{2} - x - 3}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to - \infty} \frac{4 x^{2} + 2 x - 1}{9 x^{2} - x - 3}}$

Step 2

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

$\sqrt{lim_{x \to - \infty} \frac{\frac{4 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$\sqrt{lim_{x \to - \infty} \frac{\frac{4 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.1.1.2

Divide $4$ by $1$ .

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2 x}{x^{2}} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.1.2

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

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Step 3.1.2.1

Factor $x$ out of $2 x$ .

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{x \cdot 2}{x^{2}} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

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

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{x \cdot 2}{x \cdot x} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.1.2.2.2

Cancel the common factor.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{x \cdot 2}{x \cdot x} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.1.2.2.3

Rewrite the expression.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{\frac{9 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.2.1.2

Divide $9$ by $1$ .

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.2.2

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

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Step 3.2.2.1

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

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{x^{1}}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.2.2.2

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

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{x \cdot 1}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 3.2.2.3

Cancel the common factors.

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Step 3.2.2.3.1

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

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{x \cdot 1}{x \cdot x} + \frac{- 3}{x^{2}}}}$

Step 3.2.2.3.2

Cancel the common factor.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{x \cdot 1}{x \cdot x} + \frac{- 3}{x^{2}}}}$

Step 3.2.2.3.3

Rewrite the expression.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{1}{x} + \frac{- 3}{x^{2}}}}$

Step 3.2.3

Move the negative in front of the fraction.

$\sqrt{lim_{x \to - \infty} \frac{4 + \frac{2}{x} - \frac{1}{x^{2}}}{9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 3.3

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

$\sqrt{\frac{lim_{x \to - \infty} 4 + \frac{2}{x} - \frac{1}{x^{2}}}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 3.4

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

$\sqrt{\frac{lim_{x \to - \infty} 4 + lim_{x \to - \infty} \frac{2}{x} - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

$\sqrt{\frac{4 + lim_{x \to - \infty} \frac{2}{x} - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 3.6

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

$\sqrt{\frac{4 + 2 lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 4

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

$\sqrt{\frac{4 + 2 \cdot 0 - lim_{x \to - \infty} \frac{1}{x^{2}}}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{lim_{x \to - \infty} 9 - \frac{1}{x} - \frac{3}{x^{2}}}}$

Step 6

Evaluate the limit.

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Step 6.1

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

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{lim_{x \to - \infty} 9 - lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 6.2

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

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{9 - lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 7

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

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{9 - 0 - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{9 - 0 - 3 lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\sqrt{\frac{4 + 2 \cdot 0 - 0}{9 - 0 - 3 \cdot 0}}$

Step 10

Simplify the answer.

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Step 10.1

Multiply $2$ by $0$ .

$\sqrt{\frac{4 + 0 - 0}{9 - 0 - 3 \cdot 0}}$

Step 10.2

Multiply $- 1$ by $0$ .

$\sqrt{\frac{4 + 0 + 0}{9 - 0 - 3 \cdot 0}}$

Step 10.3

Add $4 + 0$ and $0$ .

$\sqrt{\frac{4 + 0}{9 - 0 - 3 \cdot 0}}$

Step 10.4

Add $4$ and $0$ .

$\sqrt{\frac{4}{9 - 0 - 3 \cdot 0}}$

Step 10.5

Multiply $- 1$ by $0$ .

$\sqrt{\frac{4}{9 + 0 - 3 \cdot 0}}$

Step 10.6

Multiply $- 3$ by $0$ .

$\sqrt{\frac{4}{9 + 0 + 0}}$

Step 10.7

Add $9 + 0$ and $0$ .

$\sqrt{\frac{4}{9 + 0}}$

Step 10.8

Add $9$ and $0$ .

$\sqrt{\frac{4}{9}}$

Step 10.9

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$\frac{\sqrt{4}}{\sqrt{9}}$

Step 10.10

Simplify the numerator.

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Step 10.10.1

Rewrite $4$ as $2^{2}$ .

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

Step 10.10.2

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

$\frac{2}{\sqrt{9}}$

Step 10.11

Simplify the denominator.

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Step 10.11.1

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

$\frac{2}{\sqrt{3^{2}}}$

Step 10.11.2

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

$\frac{2}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$