Calculus Examples

$lim_{x \to \infty} \frac{(5 - x) (10 + 2 x)}{(3 - 8 x) (8 + 3 x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to \infty} (5 - x) (10 + 2 x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Apply the distributive property.

$\frac{lim_{x \to \infty} 5 (10 + 2 x) - x (10 + 2 x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.2

Apply the distributive property.

$\frac{lim_{x \to \infty} 5 \cdot 10 + 5 (2 x) - x (10 + 2 x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.3

Apply the distributive property.

$\frac{lim_{x \to \infty} 5 \cdot 10 + 5 (2 x) - x \cdot 10 - x (2 x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Move $x$ .

$\frac{lim_{x \to \infty} 5 \cdot 10 + 5 \cdot 2 x - 1 \cdot 10 x - x (2 x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4.2

Move $x$ .

$\frac{lim_{x \to \infty} 5 \cdot 10 + 5 \cdot 2 x - 1 \cdot 10 x - 1 \cdot 2 x \cdot x}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4.3

Multiply $5$ by $10$ .

$\frac{lim_{x \to \infty} 50 + 5 \cdot 2 x - 1 \cdot 10 x - 1 \cdot 2 x \cdot x}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4.4

Multiply $5$ by $2$ .

$\frac{lim_{x \to \infty} 50 + 10 x - 1 \cdot 10 x - 1 \cdot 2 x \cdot x}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4.5

Multiply $- 1$ by $10$ .

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 1 \cdot 2 x \cdot x}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.4.6

Multiply $- 1$ by $2$ .

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 2 x \cdot x}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.5

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

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 2 (x^{1} x)}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.6

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

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 2 (x^{1} x^{1})}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.7

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

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 2 x^{1 + 1}}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.8

Simplify by adding terms.

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Step 1.1.2.8.1

Add $1$ and $1$ .

$\frac{lim_{x \to \infty} 50 + 10 x - 10 x - 2 x^{2}}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.8.2

Subtract $10 x$ from $10 x$ .

$\frac{lim_{x \to \infty} 50 + 0 - 2 x^{2}}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.8.3

Simplify the expression.

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Step 1.1.2.8.3.1

Subtract $2 x^{2}$ from $0$ .

$\frac{lim_{x \to \infty} 50 - 2 x^{2}}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.8.3.2

Reorder $50$ and $- 2 x^{2}$ .

$\frac{lim_{x \to \infty} - 2 x^{2} + 50}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.2.9

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

$\frac{- \infty}{lim_{x \to \infty} (3 - 8 x) (8 + 3 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Apply the distributive property.

$\frac{- \infty}{lim_{x \to \infty} 3 (8 + 3 x) - 8 x (8 + 3 x)}$

Step 1.1.3.2

Apply the distributive property.

$\frac{- \infty}{lim_{x \to \infty} 3 \cdot 8 + 3 (3 x) - 8 x (8 + 3 x)}$

Step 1.1.3.3

Apply the distributive property.

$\frac{- \infty}{lim_{x \to \infty} 3 \cdot 8 + 3 (3 x) - 8 x \cdot 8 - 8 x (3 x)}$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Move $x$ .

$\frac{- \infty}{lim_{x \to \infty} 3 \cdot 8 + 3 \cdot 3 x - 8 \cdot 8 x - 8 x (3 x)}$

Step 1.1.3.4.2

Move $x$ .

$\frac{- \infty}{lim_{x \to \infty} 3 \cdot 8 + 3 \cdot 3 x - 8 \cdot 8 x - 8 \cdot 3 x \cdot x}$

Step 1.1.3.4.3

Multiply $3$ by $8$ .

$\frac{- \infty}{lim_{x \to \infty} 24 + 3 \cdot 3 x - 8 \cdot 8 x - 8 \cdot 3 x \cdot x}$

Step 1.1.3.4.4

Multiply $3$ by $3$ .

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 8 \cdot 8 x - 8 \cdot 3 x \cdot x}$

Step 1.1.3.4.5

Multiply $- 8$ by $8$ .

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 8 \cdot 3 x \cdot x}$

Step 1.1.3.4.6

Multiply $- 8$ by $3$ .

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 24 x \cdot x}$

Step 1.1.3.5

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

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 24 (x^{1} x)}$

Step 1.1.3.6

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

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 24 (x^{1} x^{1})}$

Step 1.1.3.7

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

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 24 x^{1 + 1}}$

Step 1.1.3.8

Simplify by adding terms.

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Step 1.1.3.8.1

Add $1$ and $1$ .

$\frac{- \infty}{lim_{x \to \infty} 24 + 9 x - 64 x - 24 x^{2}}$

Step 1.1.3.8.2

Subtract $64 x$ from $9 x$ .

$\frac{- \infty}{lim_{x \to \infty} 24 - 55 x - 24 x^{2}}$

Step 1.1.3.8.3

Simplify the expression.

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Step 1.1.3.8.3.1

Reorder $- 55 x$ and $- 24 x^{2}$ .

$\frac{- \infty}{lim_{x \to \infty} 24 - 24 x^{2} - 55 x}$

Step 1.1.3.8.3.2

Move $24$ .

$\frac{- \infty}{lim_{x \to \infty} - 24 x^{2} - 55 x + 24}$

Step 1.1.3.9

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

$\frac{- \infty}{- \infty}$

Step 1.1.3.10

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{- \infty}$

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{- \infty}$

Step 1.2

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.

$lim_{x \to \infty} \frac{(5 - x) (10 + 2 x)}{(3 - 8 x) (8 + 3 x)} = lim_{x \to \infty} \frac{\frac{d}{d x} [(5 - x) (10 + 2 x)]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [(5 - x) (10 + 2 x)]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.2

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 $f (x) = 5 - x$ and $g (x) = 10 + 2 x$ .

$lim_{x \to \infty} \frac{(5 - x) \frac{d}{d x} [10 + 2 x] + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.3

By the Sum Rule, the derivative of $10 + 2 x$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [2 x]$ .

$lim_{x \to \infty} \frac{(5 - x) (\frac{d}{d x} [10] + \frac{d}{d x} [2 x]) + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.4

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{(5 - x) (0 + \frac{d}{d x} [2 x]) + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.5

Add $0$ and $\frac{d}{d x} [2 x]$ .

$lim_{x \to \infty} \frac{(5 - x) \frac{d}{d x} [2 x] + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{(5 - x) (2 \frac{d}{d x} [x]) + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.7

Move $2$ to the left of $5 - x$ .

$lim_{x \to \infty} \frac{2 \cdot (5 - x) \frac{d}{d x} [x] + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{2 (5 - x) \cdot 1 + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.9

Multiply $2$ by $1$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) \frac{d}{d x} [5 - x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.10

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) (\frac{d}{d x} [5] + \frac{d}{d x} [- x])}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.11

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) (0 + \frac{d}{d x} [- x])}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.12

Add $0$ and $\frac{d}{d x} [- x]$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) \frac{d}{d x} [- x]}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.13

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) (- \frac{d}{d x} [x])}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) (- 1 \cdot 1)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.15

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{2 (5 - x) + (10 + 2 x) \cdot - 1}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.16

Move $- 1$ to the left of $10 + 2 x$ .

$lim_{x \to \infty} \frac{2 (5 - x) - 1 \cdot (10 + 2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.17

Rewrite $- 1 (10 + 2 x)$ as $- (10 + 2 x)$ .

$lim_{x \to \infty} \frac{2 (5 - x) - (10 + 2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18

Simplify.

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Step 1.3.18.1

Apply the distributive property.

$lim_{x \to \infty} \frac{2 \cdot 5 + 2 (- x) - (10 + 2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.2

Apply the distributive property.

$lim_{x \to \infty} \frac{2 \cdot 5 + 2 (- x) - 1 \cdot 10 - (2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3

Combine terms.

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Step 1.3.18.3.1

Multiply $2$ by $5$ .

$lim_{x \to \infty} \frac{10 + 2 (- x) - 1 \cdot 10 - (2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.2

Multiply $- 1$ by $2$ .

$lim_{x \to \infty} \frac{10 - 2 x - 1 \cdot 10 - (2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.3

Multiply $- 1$ by $10$ .

$lim_{x \to \infty} \frac{10 - 2 x - 10 - (2 x)}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.4

Multiply $2$ by $- 1$ .

$lim_{x \to \infty} \frac{10 - 2 x - 10 - 2 x}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.5

Subtract $10$ from $10$ .

$lim_{x \to \infty} \frac{- 2 x + 0 - 2 x}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.6

Add $- 2 x$ and $0$ .

$lim_{x \to \infty} \frac{- 2 x - 2 x}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.18.3.7

Subtract $2 x$ from $- 2 x$ .

$lim_{x \to \infty} \frac{- 4 x}{\frac{d}{d x} [(3 - 8 x) (8 + 3 x)]}$

Step 1.3.19

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 $f (x) = 3 - 8 x$ and $g (x) = 8 + 3 x$ .

$lim_{x \to \infty} \frac{- 4 x}{(3 - 8 x) \frac{d}{d x} [8 + 3 x] + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.20

By the Sum Rule, the derivative of $8 + 3 x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [3 x]$ .

$lim_{x \to \infty} \frac{- 4 x}{(3 - 8 x) (\frac{d}{d x} [8] + \frac{d}{d x} [3 x]) + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.21

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{- 4 x}{(3 - 8 x) (0 + \frac{d}{d x} [3 x]) + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.22

Add $0$ and $\frac{d}{d x} [3 x]$ .

$lim_{x \to \infty} \frac{- 4 x}{(3 - 8 x) \frac{d}{d x} [3 x] + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.23

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{- 4 x}{(3 - 8 x) (3 \frac{d}{d x} [x]) + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.24

Move $3$ to the left of $3 - 8 x$ .

$lim_{x \to \infty} \frac{- 4 x}{3 \cdot (3 - 8 x) \frac{d}{d x} [x] + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.25

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) \cdot 1 + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.26

Multiply $3$ by $1$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) \frac{d}{d x} [3 - 8 x]}$

Step 1.3.27

By the Sum Rule, the derivative of $3 - 8 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 8 x]$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) (\frac{d}{d x} [3] + \frac{d}{d x} [- 8 x])}$

Step 1.3.28

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) (0 + \frac{d}{d x} [- 8 x])}$

Step 1.3.29

Add $0$ and $\frac{d}{d x} [- 8 x]$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) \frac{d}{d x} [- 8 x]}$

Step 1.3.30

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) (- 8 \frac{d}{d x} [x])}$

Step 1.3.31

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) (- 8 \cdot 1)}$

Step 1.3.32

Multiply $- 8$ by $1$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) + (8 + 3 x) \cdot - 8}$

Step 1.3.33

Move $- 8$ to the left of $8 + 3 x$ .

$lim_{x \to \infty} \frac{- 4 x}{3 (3 - 8 x) - 8 \cdot (8 + 3 x)}$

Step 1.3.34

Simplify.

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Step 1.3.34.1

Apply the distributive property.

$lim_{x \to \infty} \frac{- 4 x}{3 \cdot 3 + 3 (- 8 x) - 8 (8 + 3 x)}$

Step 1.3.34.2

Apply the distributive property.

$lim_{x \to \infty} \frac{- 4 x}{3 \cdot 3 + 3 (- 8 x) - 8 \cdot 8 - 8 (3 x)}$

Step 1.3.34.3

Combine terms.

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Step 1.3.34.3.1

Multiply $3$ by $3$ .

$lim_{x \to \infty} \frac{- 4 x}{9 + 3 (- 8 x) - 8 \cdot 8 - 8 (3 x)}$

Step 1.3.34.3.2

Multiply $- 8$ by $3$ .

$lim_{x \to \infty} \frac{- 4 x}{9 - 24 x - 8 \cdot 8 - 8 (3 x)}$

Step 1.3.34.3.3

Multiply $- 8$ by $8$ .

$lim_{x \to \infty} \frac{- 4 x}{9 - 24 x - 64 - 8 (3 x)}$

Step 1.3.34.3.4

Multiply $3$ by $- 8$ .

$lim_{x \to \infty} \frac{- 4 x}{9 - 24 x - 64 - 24 x}$

Step 1.3.34.3.5

Subtract $64$ from $9$ .

$lim_{x \to \infty} \frac{- 4 x}{- 24 x - 55 - 24 x}$

Step 1.3.34.3.6

Subtract $24 x$ from $- 24 x$ .

$lim_{x \to \infty} \frac{- 4 x}{- 48 x - 55}$

Step 2

Move the term $- 4$ outside of the limit because it is constant with respect to $x$ .

$- 4 lim_{x \to \infty} \frac{x}{- 48 x - 55}$

Step 3

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

$- 4 lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{- 48 x}{x} + \frac{- 55}{x}}$

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

$- 4 lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{- 48 x}{x} + \frac{- 55}{x}}$

Step 4.1.2

Rewrite the expression.

$- 4 lim_{x \to \infty} \frac{1}{\frac{- 48 x}{x} + \frac{- 55}{x}}$

Step 4.2

Simplify each term.

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Step 4.2.1

Cancel the common factor of $x$ .

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Step 4.2.1.1

Cancel the common factor.

$- 4 lim_{x \to \infty} \frac{1}{\frac{- 48 x}{x} + \frac{- 55}{x}}$

Step 4.2.1.2

Divide $- 48$ by $1$ .

$- 4 lim_{x \to \infty} \frac{1}{- 48 + \frac{- 55}{x}}$

Step 4.2.2

Move the negative in front of the fraction.

$- 4 lim_{x \to \infty} \frac{1}{- 48 - \frac{55}{x}}$

Step 4.3

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

$- 4 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} - 48 - \frac{55}{x}}$

Step 4.4

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

$- 4 \frac{1}{lim_{x \to \infty} - 48 - \frac{55}{x}}$

Step 4.5

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

$- 4 \frac{1}{- lim_{x \to \infty} 48 - lim_{x \to \infty} \frac{55}{x}}$

Step 4.6

Evaluate the limit of $48$ which is constant as $x$ approaches $\infty$ .

$- 4 \frac{1}{- 1 \cdot 48 - lim_{x \to \infty} \frac{55}{x}}$

Step 4.7

Move the term $55$ outside of the limit because it is constant with respect to $x$ .

$- 4 \frac{1}{- 48 - 55 lim_{x \to \infty} \frac{1}{x}}$

Step 5

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

$- 4 \frac{1}{- 48 - 55 \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Multiply $- 55$ by $0$ .

$- 4 \frac{1}{- 48 + 0}$

Step 6.1.2

Add $- 48$ and $0$ .

$- 4 (\frac{1}{- 48})$

Step 6.2

Cancel the common factor of $4$ .

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Step 6.2.1

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{1}{- 48}$

Step 6.2.2

Factor $4$ out of $- 48$ .

$4 \cdot - 1 \frac{1}{4 \cdot - 12}$

Step 6.2.3

Cancel the common factor.

$4 \cdot - 1 \frac{1}{4 \cdot - 12}$

Step 6.2.4

Rewrite the expression.

$- \frac{1}{- 12}$

Step 6.3

Move the negative in front of the fraction.

$- - \frac{1}{12}$

Step 6.4

Multiply $- - \frac{1}{12}$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{12})$

Step 6.4.2

Multiply $\frac{1}{12}$ by $1$ .

$\frac{1}{12}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{12}$

Decimal Form:

$0.08\overline{3}$