Calculus Examples

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{10} + 8 x^{7}}}$

Step 1

Simplify.

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Step 1.1

Factor $x^{7}$ out of $x^{10} + 8 x^{7}$ .

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Step 1.1.1

Factor $x^{7}$ out of $x^{10}$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} x^{3} + 8 x^{7}}}$

Step 1.1.2

Factor $x^{7}$ out of $8 x^{7}$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} x^{3} + x^{7} \cdot 8}}$

Step 1.1.3

Factor $x^{7}$ out of $x^{7} x^{3} + x^{7} \cdot 8$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} (x^{3} + 8)}}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} (x^{3} + 2^{3})}}$

Step 1.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} ((x + 2) (x^{2} - x \cdot 2 + 2^{2}))}}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $2$ by $- 1$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} ((x + 2) (x^{2} - 2 x + 2^{2}))}}$

Step 1.4.2

Raise $2$ to the power of $2$ .

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} ((x + 2) (x^{2} - 2 x + 4))}}$

$lim_{x \to - \infty} \frac{x^{5} + 4 x^{2}}{\sqrt{x^{7} (x + 2) (x^{2} - 2 x + 4)}}$

Step 1.5

Rewrite $x^{7} (x + 2) (x^{2} - 2 x + 4)$ as ${(x^{3})}^{2} (x (x + 2) (x^{2} - 2 x + 2^{2}))$ .

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Step 1.5.1

Factor out $x^{6}$ .

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

Step 1.5.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

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

Step 1.5.3

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

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

Step 1.5.4

Add parentheses.

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

Step 1.5.5

Add parentheses.

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

Step 1.6

Pull terms out from under the radical.

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

Step 1.7

Raise $2$ to the power of $2$ .

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{5}$ .

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $x^{5}$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{1 + \frac{4 x^{2}}{x^{5}}}{\frac{x^{3} \sqrt{x (x + 2) (x^{2} - 2 x + 4)}}{x^{5}}}$

Step 3.1.2

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 3.1.2.1

Factor $x^{2}$ out of $4 x^{2}$ .

$lim_{x \to - \infty} \frac{1 + \frac{x^{2} \cdot 4}{x^{5}}}{\frac{x^{3} \sqrt{x (x + 2) (x^{2} - 2 x + 4)}}{x^{5}}}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

Factor $x^{2}$ out of $x^{5}$ .

$lim_{x \to - \infty} \frac{1 + \frac{x^{2} \cdot 4}{x^{2} x^{3}}}{\frac{x^{3} \sqrt{x (x + 2) (x^{2} - 2 x + 4)}}{x^{5}}}$

Step 3.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{1 + \frac{x^{2} \cdot 4}{x^{2} x^{3}}}{\frac{x^{3} \sqrt{x (x + 2) (x^{2} - 2 x + 4)}}{x^{5}}}$

Step 3.1.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{1 + \frac{4}{x^{3}}}{\frac{x^{3} \sqrt{x (x + 2) (x^{2} - 2 x + 4)}}{x^{5}}}$

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $x$ by $x$ .

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

Step 3.2.3.2

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

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

Step 4

Expand $(x^{2} + 2 x) (x^{2} - 2 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5

Evaluate the limit.

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Step 5.1

Simplify terms.

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Step 5.1.1

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 4 + 2 x \cdot x^{2} + 2 x (- 2 x) + 2 x \cdot 4$ .

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Step 5.1.1.1

Reorder the factors in the terms $x^{2} (- 2 x)$ and $2 x \cdot x^{2}$ .

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

Step 5.1.1.2

Add $- 2 x^{2} x$ and $2 x^{2} x$ .

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

Step 5.1.1.3

Add $x^{2} x^{2} + x^{2} \cdot 4$ and $0$ .

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

Step 5.1.2

Simplify each term.

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Step 5.1.2.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.1.2.1.1

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

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

Step 5.1.2.1.2

Add $2$ and $2$ .

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

Step 5.1.2.2

Move $4$ to the left of $x^{2}$ .

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

Step 5.1.2.3

Rewrite using the commutative property of multiplication.

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

Step 5.1.2.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.2.4.1

Move $x$ .

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

Step 5.1.2.4.2

Multiply $x$ by $x$ .

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

Step 5.1.2.5

Multiply $2$ by $- 2$ .

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

Step 5.1.2.6

Multiply $4$ by $2$ .

$lim_{x \to - \infty} \frac{1 + \frac{4}{x^{3}}}{\frac{\sqrt{x^{4} + 4 x^{2} - 4 x^{2} + 8 x}}{x^{2}}}$

Step 5.1.3

Combine the opposite terms in $x^{4} + 4 x^{2} - 4 x^{2} + 8 x$ .

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Step 5.1.3.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$lim_{x \to - \infty} \frac{1 + \frac{4}{x^{3}}}{\frac{\sqrt{x^{4} + 0 + 8 x}}{x^{2}}}$

Step 5.1.3.2

Add $x^{4}$ and $0$ .

$lim_{x \to - \infty} \frac{1 + \frac{4}{x^{3}}}{\frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 5.2

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

$\frac{lim_{x \to - \infty} 1 + \frac{4}{x^{3}}}{lim_{x \to - \infty} \frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 5.3

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

$\frac{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} \frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 5.4

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

$\frac{1 + lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} \frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 5.5

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

$\frac{1 + 4 lim_{x \to - \infty} \frac{1}{x^{3}}}{lim_{x \to - \infty} \frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{1 + 4 \cdot 0}{lim_{x \to - \infty} \frac{\sqrt{x^{4} + 8 x}}{x^{2}}}$

Step 7

Simplify.

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Step 7.1

Factor $x$ out of $x^{4} + 8 x$ .

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Step 7.1.1

Factor $x$ out of $x^{4}$ .

$\frac{1 + 4 \cdot 0}{lim_{x \to - \infty} \frac{\sqrt{x \cdot x^{3} + 8 x}}{x^{2}}}$

Step 7.1.2

Factor $x$ out of $8 x$ .

$\frac{1 + 4 \cdot 0}{lim_{x \to - \infty} \frac{\sqrt{x \cdot x^{3} + x \cdot 8}}{x^{2}}}$

Step 7.1.3

Factor $x$ out of $x \cdot x^{3} + x \cdot 8$ .

$\frac{1 + 4 \cdot 0}{lim_{x \to - \infty} \frac{\sqrt{x (x^{3} + 8)}}{x^{2}}}$

Step 7.2

Rewrite $8$ as $2^{3}$ .

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

Step 7.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 7.4

Simplify.

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Step 7.4.1

Multiply $2$ by $- 1$ .

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

Step 7.4.2

Raise $2$ to the power of $2$ .

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

Step 8

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $- \sqrt{x^{4}} = x^{2}$ .

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

Step 9

Evaluate the limit.

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Step 9.1

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

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $x$ out of $x^{4}$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 9.3

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

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

Step 9.4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 9.5

Move the limit under the radical sign.

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

Step 10

Apply L'Hospital's rule.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

Evaluate the limit of the numerator.

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Step 10.1.2.1

Apply the distributive property.

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

Step 10.1.2.2

Apply the distributive property.

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

Step 10.1.2.3

Apply the distributive property.

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

Step 10.1.2.4

Apply the distributive property.

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

Step 10.1.2.5

Apply the distributive property.

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

Step 10.1.2.6

Simplify with commuting.

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Step 10.1.2.6.1

Reorder $x$ and $- 2$ .

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

Step 10.1.2.6.2

Reorder $x$ and $4$ .

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

Step 10.1.2.7

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

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

Step 10.1.2.8

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

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

Step 10.1.2.9

Add $1$ and $2$ .

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

Step 10.1.2.10

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

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

Step 10.1.2.11

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

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

Step 10.1.2.12

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

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

Step 10.1.2.13

Simplify by adding terms.

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Step 10.1.2.13.1

Add $1$ and $1$ .

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

Step 10.1.2.13.2

Simplify the expression.

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Step 10.1.2.13.2.1

Multiply $2$ by $- 2$ .

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

Step 10.1.2.13.2.2

Multiply $2$ by $4$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} - 2 x^{2} + 4 x + 2 x^{2} - 4 x + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.2.3

Move $4 x$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} - 2 x^{2} + 2 x^{2} + 4 x - 4 x + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.2.4

Move $- 2 x^{2}$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} + 2 x^{2} - 2 x^{2} + 4 x - 4 x + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.3

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} + 0 + 4 x - 4 x + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.4

Add $x^{3}$ and $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} + 4 x - 4 x + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.5

Subtract $4 x$ from $4 x$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} + 0 + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.13.6

Add $x^{3}$ and $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{\frac{lim_{x \to - \infty} x^{3} + 8}{lim_{x \to - \infty} x^{3}}}}{lim_{x \to - \infty} 1}}$

Step 10.1.2.14

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

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

Step 10.1.3

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

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

Step 10.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 10.2

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.

$lim_{x \to - \infty} \frac{(x + 2) (x^{2} - 2 x + 4)}{x^{3}} = lim_{x \to - \infty} \frac{\frac{d}{d x} [(x + 2) (x^{2} - 2 x + 4)]}{\frac{d}{d x} [x^{3}]}$

Step 10.3

Find the derivative of the numerator and denominator.

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Step 10.3.1

Differentiate the numerator and denominator.

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{d}{d x} [(x + 2) (x^{2} - 2 x + 4)]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.2

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 $f (x) = x + 2$ and $g (x) = x^{2} - 2 x + 4$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) \frac{d}{d x} [x^{2} - 2 x + 4] + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.3

By the Sum Rule, the derivative of $x^{2} - 2 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [4]$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [4]) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [4]) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.5

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.6

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2 \cdot 1 + \frac{d}{d x} [4]) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.7

Multiply $- 2$ by $1$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2 + \frac{d}{d x} [4]) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.8

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2 + 0) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.9

Add $2 x - 2$ and $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + (x^{2} - 2 x + 4) \frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.10

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + (x^{2} - 2 x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.11

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + (x^{2} - 2 x + 4) (1 + \frac{d}{d x} [2])}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.12

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + (x^{2} - 2 x + 4) (1 + 0)}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.13

Add $1$ and $0$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + (x^{2} - 2 x + 4) \cdot 1}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.14

Multiply $x^{2} - 2 x + 4$ by $1$ .

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) (2 x - 2) + x^{2} - 2 x + 4}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.15

Simplify.

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Step 10.3.15.1

Apply the distributive property.

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{(x + 2) \cdot 2 x + (x + 2) \cdot - 2 + x^{2} - 2 x + 4}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.15.2

Apply the distributive property.

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{lim_{x \to - \infty} \frac{x \cdot 2 x + 2 \cdot 2 x + (x + 2) \cdot - 2 + x^{2} - 2 x + 4}{\frac{d}{d x} [x^{3}]}}}{lim_{x \to - \infty} 1}}$

Step 10.3.15.3

Apply the distributive property.

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

Step 10.3.15.4

Combine terms.

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Step 10.3.15.4.1

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

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

Step 10.3.15.4.2

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

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

Step 10.3.15.4.3

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

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

Step 10.3.15.4.4

Add $1$ and $1$ .

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

Step 10.3.15.4.5

Multiply $2$ by $2$ .

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

Step 10.3.15.4.6

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

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

Step 10.3.15.4.7

Multiply $2$ by $- 2$ .

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

Step 10.3.15.4.8

Subtract $2 x$ from $4 x$ .

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

Step 10.3.15.4.9

Add $2 x^{2}$ and $x^{2}$ .

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

Step 10.3.15.4.10

Subtract $2 x$ from $2 x$ .

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

Step 10.3.15.4.11

Add $3 x^{2}$ and $0$ .

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

Step 10.3.15.4.12

Add $- 4$ and $4$ .

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

Step 10.3.15.4.13

Add $3 x^{2}$ and $0$ .

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

Step 10.3.16

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 10.4

Reduce.

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Step 10.4.1

Cancel the common factor of $3$ .

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Step 10.4.1.1

Cancel the common factor.

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

Step 10.4.1.2

Rewrite the expression.

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

Step 10.4.2

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

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Step 10.4.2.1

Cancel the common factor.

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

Step 10.4.2.2

Rewrite the expression.

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

Step 11

Evaluate the limit.

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Step 11.1

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

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

Step 11.2

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

$\frac{1 + 4 \cdot 0}{\frac{- \sqrt{1}}{1}}$

Step 11.3

Simplify the answer.

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Step 11.3.1

Divide $- \sqrt{1}$ by $1$ .

$\frac{1 + 4 \cdot 0}{- \sqrt{1}}$

Step 11.3.2

Simplify the numerator.

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Step 11.3.2.1

Multiply $4$ by $0$ .

$\frac{1 + 0}{- \sqrt{1}}$

Step 11.3.2.2

Add $1$ and $0$ .

$\frac{1}{- \sqrt{1}}$

Step 11.3.3

Any root of $1$ is $1$ .

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

Step 11.3.4

Multiply $- 1$ by $1$ .

$\frac{1}{- 1}$

Step 11.3.5

Divide $1$ by $- 1$ .

$- 1$