Calculus Examples

lim x \to - \infty \frac{\sqrt{9 x^{2} - 4}}{3 x + 5}

$lim_{x \to - \infty} \frac{\sqrt{9 x^{2} - 4}}{3 x + 5}$

Step 1

Simplify.

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Step 1.1

Rewrite

9 x^{2}

$9 x^{2}$ as

{(3 x)}^{2}

${(3 x)}^{2}$ .

lim x \to - \infty \frac{\sqrt{{(3 x)}^{2} - 4}}{3 x + 5}

$lim_{x \to - \infty} \frac{\sqrt{{(3 x)}^{2} - 4}}{3 x + 5}$

Step 1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

lim x \to - \infty \frac{\sqrt{{(3 x)}^{2} - 2^{2}}}{3 x + 5}

$lim_{x \to - \infty} \frac{\sqrt{{(3 x)}^{2} - 2^{2}}}{3 x + 5}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 3 x

$a = 3 x$ and

b = 2

$b = 2$ .

lim x \to - \infty \frac{\sqrt{(3 x + 2) (3 x - 2)}}{3 x + 5}

$lim_{x \to - \infty} \frac{\sqrt{(3 x + 2) (3 x - 2)}}{3 x + 5}$

lim x \to - \infty \frac{\sqrt{(3 x + 2) (3 x - 2)}}{3 x + 5}

$lim_{x \to - \infty} \frac{\sqrt{(3 x + 2) (3 x - 2)}}{3 x + 5}$

Step 2

Divide the numerator and denominator by the highest power of

x

$x$ in the denominator, which is

- \sqrt{x^{2}} = x

$- \sqrt{x^{2}} = x$ .

lim x \to - \infty \frac{- \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{\frac{3 x}{x} + \frac{5}{x}}

$lim_{x \to - \infty} \frac{- \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{\frac{3 x}{x} + \frac{5}{x}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of

x

$x$ .

lim x \to - \infty \frac{- \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{3 + \frac{5}{x}}

$lim_{x \to - \infty} \frac{- \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{3 + \frac{5}{x}}$

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 - \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{lim x \to - \infty 3 + \frac{5}{x}}

$\frac{lim_{x \to - \infty} - \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 3.3

Move the term

$- 1$ outside of the limit because it is constant with respect to

$x$ .

$\frac{- lim_{x \to - \infty} \sqrt{\frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 3.4

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (3 x - 2)}{x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 4.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{- \sqrt{\frac{lim_{x \to - \infty} (3 x + 2) (3 x - 2)}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

Apply the distributive property.

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 3 x (3 x - 2) + 2 (3 x - 2)}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.2

Apply the distributive property.

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 3 x \cdot 3 x + 3 x \cdot - 2 + 2 (3 x - 2)}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.3

Apply the distributive property.

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 3 x \cdot 3 x + 3 x \cdot - 2 + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.4

Simplify the expression.

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Step 4.1.2.4.1

Move

$x$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 3 \cdot 3 x \cdot x + 3 x \cdot - 2 + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.4.2

Move

$x$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 3 \cdot 3 x \cdot x + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.4.3

Multiply

$3$ by

$3$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x \cdot x + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.5

Raise

$x$ to the power of

$1$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{1} x + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.6

Raise

$x$ to the power of

$1$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{1} x^{1} + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{1 + 1} + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8

Simplify by adding terms.

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Step 4.1.2.8.1

Add

$1$ and

$1$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} + 3 \cdot - 2 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8.2

Multiply.

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Step 4.1.2.8.2.1

Multiply

$3$ by

$- 2$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} - 6 x + 2 \cdot 3 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8.2.2

Multiply

$2$ by

$3$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} - 6 x + 6 x + 2 \cdot - 2}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8.2.3

Multiply

$2$ by

$- 2$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} - 6 x + 6 x - 4}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8.3

Add

$- 6 x$ and

$6 x$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} + 0 - 4}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.8.4

Subtract

$4$ from

$0$ .

$\frac{- \sqrt{\frac{lim_{x \to - \infty} 9 x^{2} - 4}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.2.9

The limit at negative infinity of a polynomial of even degree whose leading coefficient is positive is infinity.

$\frac{- \sqrt{\frac{\infty}{lim_{x \to - \infty} x^{2}}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.1.3

The limit at negative infinity of a polynomial of even degree whose leading coefficient is positive is infinity.

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

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 4.2

Since

$\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to - \infty} \frac{(3 x + 2) (3 x - 2)}{x^{2}} = lim_{x \to - \infty} \frac{\frac{d}{d x} [(3 x + 2) (3 x - 2)]}{\frac{d}{d x} [x^{2}]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{d}{d x} [(3 x + 2) (3 x - 2)]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.2

Differentiate using the Product Rule which states that

$\frac{d}{d x} [f (x) g (x)]$ is

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

$f (x) = 3 x + 2$ and

$g (x) = 3 x - 2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) \frac{d}{d x} [3 x - 2] + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.3

By the Sum Rule, the derivative of

$3 x - 2$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.4

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.5

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (3 \cdot 1 + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.6

Multiply

$3$ by

$1$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (3 + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.7

Since

$- 2$ is constant with respect to

$x$ , the derivative of

$- 2$ with respect to

$x$ is

$0$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) (3 + 0) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.8

Add

$3$ and

$0$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{(3 x + 2) \cdot 3 + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.9

Move

$3$ to the left of

$3 x + 2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 \cdot (3 x + 2) + (3 x - 2) \frac{d}{d x} [3 x + 2]}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.10

By the Sum Rule, the derivative of

$3 x + 2$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) (\frac{d}{d x} [3 x] + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.11

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) (3 \frac{d}{d x} [x] + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.12

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) (3 \cdot 1 + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.13

Multiply

$3$ by

$1$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) (3 + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.14

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) (3 + 0)}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.15

Add

$3$ and

$0$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + (3 x - 2) \cdot 3}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.16

Move

$3$ to the left of

$3 x - 2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 (3 x + 2) + 3 \cdot (3 x - 2)}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17

Simplify.

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Step 4.3.17.1

Apply the distributive property.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 \cdot 3 x + 3 \cdot 2 + 3 (3 x - 2)}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.2

Apply the distributive property.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{3 \cdot 3 x + 3 \cdot 2 + 3 \cdot 3 x + 3 \cdot - 2}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3

Combine terms.

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Step 4.3.17.3.1

Multiply

$3$ by

$3$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{9 x + 3 \cdot 2 + 3 \cdot 3 x + 3 \cdot - 2}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.2

Multiply

$3$ by

$2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{9 x + 6 + 3 \cdot 3 x + 3 \cdot - 2}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.3

Multiply

$3$ by

$3$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{9 x + 6 + 9 x + 3 \cdot - 2}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.4

Multiply

$3$ by

$- 2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{9 x + 6 + 9 x - 6}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.5

Add

$9 x$ and

$9 x$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{18 x + 6 - 6}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.6

Subtract

$6$ from

$6$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{18 x + 0}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.17.3.7

Add

$18 x$ and

$0$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{18 x}{\frac{d}{d x} [x^{2}]}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.3.18

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{18 x}{2 x}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.4

Reduce.

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Step 4.4.1

Cancel the common factor of

$18$ and

$2$ .

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Step 4.4.1.1

Factor

$2$ out of

$18 x$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 9 x}{2 x}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.4.1.2

Cancel the common factors.

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Step 4.4.1.2.1

Factor

$2$ out of

$2 x$ .

$\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 9 x}{2 (x)}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.4.1.2.2

Cancel the common factor.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 9 x}{2 x}}}{lim_{x \to - \infty} 3 + \frac{5}{x}}$

Step 4.4.1.2.3

Rewrite the expression.

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

Step 4.4.2

Cancel the common factor of

$x$ .

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Divide

$9$ by

$1$ .

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

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit of

$9$ which is constant as

$x$ approaches

$- \infty$ .

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

Step 5.2

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

$x$ approaches

$- \infty$ .

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

Step 5.3

Evaluate the limit of

$3$ which is constant as

$x$ approaches

$- \infty$ .

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

Step 5.4

Move the term

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

$x$ .

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

Step 6

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

$\frac{1}{x}$ approaches

$0$ .

$\frac{- \sqrt{9}}{3 + 5 \cdot 0}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Rewrite

$9$ as

$3^{2}$ .

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

Step 7.1.2

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

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

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply

$5$ by

$0$ .

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

Step 7.2.2

Add

$3$ and

$0$ .

$\frac{- 1 \cdot 3}{3}$

Step 7.3

Multiply

$- 1$ by

$3$ .

$\frac{- 3}{3}$

Step 7.4

Divide

$- 3$ by

$3$ .

$- 1$