Calculus Examples

$lim_{x \to - \infty} \frac{\sqrt{16 x^{6} - x^{2}}}{6 x^{3} + x^{2}}$

Step 1

Simplify.

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Step 1.1

Factor $x^{2}$ out of $16 x^{6} - x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $16 x^{6}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} (16 x^{4}) - x^{2}}}{6 x^{3} + x^{2}}$

Step 1.1.2

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

$lim_{x \to - \infty} \frac{\sqrt{x^{2} (16 x^{4}) + x^{2} \cdot - 1}}{6 x^{3} + x^{2}}$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} (16 x^{4}) + x^{2} \cdot - 1$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} (16 x^{4} - 1)}}{6 x^{3} + x^{2}}$

Step 1.2

Rewrite $16 x^{4}$ as ${(4 x^{2})}^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ({(4 x^{2})}^{2} - 1)}}{6 x^{3} + x^{2}}$

Step 1.3

Rewrite $1$ as $1^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ({(4 x^{2})}^{2} - 1^{2})}}{6 x^{3} + x^{2}}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x^{2}$ and $b = 1$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ((4 x^{2} + 1) (4 x^{2} - 1))}}{6 x^{3} + x^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ((4 x^{2} + 1) ({(2 x)}^{2} - 1))}}{6 x^{3} + x^{2}}$

Step 1.5.2

Rewrite $1$ as $1^{2}$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ((4 x^{2} + 1) ({(2 x)}^{2} - 1^{2}))}}{6 x^{3} + x^{2}}$

Step 1.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$lim_{x \to - \infty} \frac{\sqrt{x^{2} ((4 x^{2} + 1) (2 x + 1) (2 x - 1))}}{6 x^{3} + x^{2}}$

$lim_{x \to - \infty} \frac{\sqrt{x^{2} (4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{6 x^{3} + x^{2}}$

Step 1.6

Rewrite $x^{2} (4 x^{2} + 1) (2 x + 1) (2 x - 1)$ as $x^{2} (({(2 x)}^{2} + 1) (2 x + 1) (2 x - 1))$ .

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Step 1.6.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 1.6.2

Add parentheses.

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

Step 1.6.3

Add parentheses.

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

Step 1.7

Pull terms out from under the radical.

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

Step 1.8

Apply the product rule to $2 x$ .

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

Step 1.9

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

$lim_{x \to - \infty} \frac{x \sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{6 x^{3} + x^{2}}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$lim_{x \to - \infty} \frac{\frac{x \sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{x^{3}}}{\frac{6 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}}}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Factor $x$ out of $x \sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}$ .

$lim_{x \to - \infty} \frac{\frac{x (\sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)})}{x^{3}}}{\frac{6 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{\frac{x (\sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)})}{x \cdot x^{2}}}{\frac{6 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}}}$

Step 3.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{x \sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{x \cdot x^{2}}}{\frac{6 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}}}$

Step 3.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{x^{2}}}{\frac{6 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}}}$

Step 3.2

Simplify each term.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 4

Expand $(4 x^{2} + 1) (2 x + 1)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 x^{2} (2 x + 1) + 1 (2 x + 1)) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 4.2

Apply the distributive property.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 x^{2} (2 x) + 4 x^{2} \cdot 1 + 1 (2 x + 1)) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 4.3

Apply the distributive property.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 x^{2} (2 x) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5

Simplify each term.

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Step 5.1

Rewrite using the commutative property of multiplication.

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 \cdot 2 x^{2} x + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.2

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

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Step 5.2.1

Move $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 \cdot 2 (x \cdot x^{2}) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.2.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.2.1

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 \cdot 2 (x^{1} x^{2}) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.2.2.2

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 \cdot 2 x^{1 + 2} + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.2.3

Add $1$ and $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(4 \cdot 2 x^{3} + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.3

Multiply $4$ by $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(8 x^{3} + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.4

Multiply $4$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(8 x^{3} + 4 x^{2} + 1 (2 x) + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.5

Multiply $2 x$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(8 x^{3} + 4 x^{2} + 2 x + 1 \cdot 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 5.6

Multiply $1$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{(8 x^{3} + 4 x^{2} + 2 x + 1) (2 x - 1)}}{x^{2}}}{6 + \frac{1}{x}}$

Step 6

Expand $(8 x^{3} + 4 x^{2} + 2 x + 1) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 2 x \cdot - 1 + 1 (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7

Simplify terms.

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Step 7.1

Combine the opposite terms in $8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 2 x \cdot - 1 + 1 (2 x) + 1 \cdot - 1$ .

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Step 7.1.1

Reorder the factors in the terms $2 x \cdot - 1$ and $1 (2 x)$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) - 1 \cdot 2 x + 1 \cdot 2 x + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.1.2

Add $- 1 \cdot 2 x$ and $1 \cdot 2 x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 0 + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.1.3

Add $8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x)$ and $0$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 x^{3} (2 x) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2

Simplify each term.

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Step 7.2.1

Rewrite using the commutative property of multiplication.

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 \cdot 2 x^{3} x + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 7.2.2.1

Move $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 \cdot 2 (x \cdot x^{3}) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.2.2

Multiply $x$ by $x^{3}$ .

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Step 7.2.2.2.1

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 \cdot 2 (x^{1} x^{3}) + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.2.2.2

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 \cdot 2 x^{1 + 3} + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.2.3

Add $1$ and $3$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{8 \cdot 2 x^{4} + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.3

Multiply $8$ by $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} + 8 x^{3} \cdot - 1 + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.4

Multiply $- 1$ by $8$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 x^{2} (2 x) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.5

Rewrite using the commutative property of multiplication.

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 \cdot 2 x^{2} x + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.6

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

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Step 7.2.6.1

Move $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 \cdot 2 (x \cdot x^{2}) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.6.2

Multiply $x$ by $x^{2}$ .

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Step 7.2.6.2.1

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 \cdot 2 (x^{1} x^{2}) + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.6.2.2

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

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 \cdot 2 x^{1 + 2} + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.6.3

Add $1$ and $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 4 \cdot 2 x^{3} + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.7

Multiply $4$ by $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} + 4 x^{2} \cdot - 1 + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.8

Multiply $- 1$ by $4$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 2 x (2 x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.9

Rewrite using the commutative property of multiplication.

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 2 \cdot 2 x \cdot x + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.10

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

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Step 7.2.10.1

Move $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 2 \cdot 2 (x \cdot x) + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.10.2

Multiply $x$ by $x$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 2 \cdot 2 x^{2} + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.11

Multiply $2$ by $2$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 4 x^{2} + 1 \cdot - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.2.12

Multiply $- 1$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 4 x^{2} - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.3

Combine the opposite terms in $16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 4 x^{2} - 1$ .

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Step 7.3.1

Add $- 8 x^{3}$ and $8 x^{3}$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} + 0 - 4 x^{2} + 4 x^{2} - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.3.2

Add $16 x^{4}$ and $0$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 4 x^{2} + 4 x^{2} - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.3.3

Add $- 4 x^{2}$ and $4 x^{2}$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} + 0 - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 7.3.4

Add $16 x^{4}$ and $0$ .

$lim_{x \to - \infty} \frac{\frac{\sqrt{16 x^{4} - 1}}{x^{2}}}{6 + \frac{1}{x}}$

Step 8

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

$\frac{lim_{x \to - \infty} \frac{\sqrt{16 x^{4} - 1}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9

Simplify.

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Step 9.1

Rewrite $16 x^{4}$ as ${(4 x^{2})}^{2}$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{{(4 x^{2})}^{2} - 1}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9.2

Rewrite $1$ as $1^{2}$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{{(4 x^{2})}^{2} - 1^{2}}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x^{2}$ and $b = 1$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{(4 x^{2} + 1) (4 x^{2} - 1)}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9.4

Simplify.

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Step 9.4.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{(4 x^{2} + 1) ({(2 x)}^{2} - 1)}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{(4 x^{2} + 1) ({(2 x)}^{2} - 1^{2})}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 9.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$\frac{lim_{x \to - \infty} \frac{\sqrt{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}}{x^{2}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 10

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

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{x^{4}}}}{\frac{x^{2}}{x^{2}}}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 11

Evaluate the limit.

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Step 11.1

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

$\frac{lim_{x \to - \infty} \frac{- \sqrt{\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{x^{4}}}}{1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 11.2

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

$\frac{\frac{lim_{x \to - \infty} - \sqrt{\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 11.3

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

$\frac{\frac{- lim_{x \to - \infty} \sqrt{\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 11.4

Move the limit under the radical sign.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12

Apply L'Hospital's rule.

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Step 12.1

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

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Step 12.1.1

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} (4 x^{2} + 1) (2 x + 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2

Evaluate the limit of the numerator.

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Step 12.1.2.1

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} (4 x^{2} (2 x + 1) + 1 (2 x + 1)) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.2

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} (4 x^{2} \cdot 2 x + 4 x^{2} \cdot 1 + 1 (2 x + 1)) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.3

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} (4 x^{2} \cdot 2 x + 4 x^{2} \cdot 1 + 1 \cdot 2 x + 1 \cdot 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.4

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} (4 x^{2} \cdot 2 x + 4 x^{2} \cdot 1) (2 x - 1) + (1 \cdot 2 x + 1 \cdot 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.5

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x (2 x - 1) + 4 x^{2} \cdot 1 (2 x - 1) + (1 \cdot 2 x + 1 \cdot 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.6

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 (2 x - 1) + (1 \cdot 2 x + 1 \cdot 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.7

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + (1 \cdot 2 x + 1 \cdot 1) (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.8

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x (2 x - 1) + 1 \cdot 1 (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.9

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 (2 x - 1)}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.10

Apply the distributive property.

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 x^{2} \cdot 2 x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11

Simplify the expression.

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Step 12.1.2.11.1

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 x^{2} x \cdot 2 x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.2

Move $x$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 x^{2} \cdot 2 x \cdot x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.3

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 x^{2} \cdot 2 x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.4

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 x^{2} x \cdot - 1 + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.5

Move $x$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 x^{2} \cdot - 1 x + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.6

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 x^{2} \cdot 1 \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.7

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 x^{2} \cdot 2 x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.8

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 x^{2} \cdot 1 \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.9

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 x^{2} \cdot - 1 + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.10

Move $x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 x \cdot 2 x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.11

Move $x$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 x \cdot - 1 + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.12

Move $x$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 4 \cdot 2 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.13

Multiply $4$ by $2$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 8 \cdot 2 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.11.14

Multiply $8$ by $2$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{2} x \cdot x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.12

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{2} x^{1} x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.13

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{2 + 1} x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.14

Add $2$ and $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{3} x + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.15

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{3} x^{1} + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.16

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{3 + 1} + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.17

Add $3$ and $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} + 4 \cdot 2 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.18

Multiply $4$ by $2$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} + 8 \cdot - 1 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.19

Multiply $8$ by $- 1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{2} x + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.20

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{2} x^{1} + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.21

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{2 + 1} + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.22

Add $2$ and $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 4 \cdot 1 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.23

Multiply $4$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 4 \cdot 2 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.24

Multiply $4$ by $2$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{2} x + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.25

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{2} x^{1} + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.26

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{2 + 1} + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.27

Simplify the expression.

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Step 12.1.2.27.1

Add $2$ and $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{3} + 4 \cdot 1 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.27.2

Multiply $4$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{3} + 4 \cdot - 1 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.27.3

Multiply $4$ by $- 1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 8 x^{3} + 8 x^{3} - 4 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.28

Add $- 8 x^{3}$ and $8 x^{3}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} + 0 - 4 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.29

Simplify the expression.

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Step 12.1.2.29.1

Subtract $4 x^{2}$ from $0$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 1 \cdot 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.29.2

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 2 \cdot 2 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.29.3

Multiply $2$ by $2$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x \cdot x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.30

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{1} x + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.31

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{1} x^{1} + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.32

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{1 + 1} + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33

Simplify by adding terms.

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Step 12.1.2.33.1

Add $1$ and $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} + 1 \cdot 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2

Multiply.

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Step 12.1.2.33.2.1

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} + 2 \cdot - 1 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2.2

Multiply $2$ by $- 1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 2 x + 1 \cdot 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2.3

Simplify.

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Step 12.1.2.33.2.3.1

Multiply $1$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 2 x + 1 \cdot 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2.3.2

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 2 x + 2 x + 1 \cdot 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2.3.3

Multiply $1$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 2 x + 2 x + 1 \cdot - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.2.3.4

Multiply $- 1$ by $1$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 2 x + 2 x - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.3

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

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} + 0 - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.4

Subtract $1$ from $0$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 4 x^{2} + 4 x^{2} - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.5

Add $- 4 x^{2}$ and $4 x^{2}$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} + 0 - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.33.6

Subtract $1$ from $0$ .

$\frac{\frac{- \sqrt{\frac{lim_{x \to - \infty} 16 x^{4} - 1}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.2.34

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

$\frac{\frac{- \sqrt{\frac{\infty}{lim_{x \to - \infty} x^{4}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.1.3

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

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

Step 12.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 12.2

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.

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

Step 12.3

Find the derivative of the numerator and denominator.

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Step 12.3.1

Differentiate the numerator and denominator.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{\frac{d}{d x} [(4 x^{2} + 1) (2 x + 1) (2 x - 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.2

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 $f (x) = (4 x^{2} + 1) (2 x + 1)$ and $g (x) = 2 x - 1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) \frac{d}{d x} [2 x - 1] + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.3

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.4

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.5

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (2 \cdot 1 + \frac{d}{d x} [- 1]) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.6

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (2 + \frac{d}{d x} [- 1]) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) (2 + 0) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.8

Add $2$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(4 x^{2} + 1) (2 x + 1) \cdot 2 + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.9

Move $2$ to the left of $(4 x^{2} + 1) (2 x + 1)$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot (4 x^{2} + 1) (2 x + 1) + (2 x - 1) \frac{d}{d x} [(4 x^{2} + 1) (2 x + 1)]}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.10

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 $f (x) = 4 x^{2} + 1$ and $g (x) = 2 x + 1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) \frac{d}{d x} [2 x + 1] + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.11

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.12

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.13

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) (2 \cdot 1 + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.14

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) (2 + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.15

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) (2 + 0) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.16

Add $2$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) ((4 x^{2} + 1) \cdot 2 + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.17

Move $2$ to the left of $4 x^{2} + 1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 \cdot (4 x^{2} + 1) + (2 x + 1) \frac{d}{d x} [4 x^{2} + 1])}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.18

By the Sum Rule, the derivative of $4 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [1]))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.19

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.20

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) (4 \cdot 2 x + \frac{d}{d x} [1]))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.21

Multiply $2$ by $4$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) (8 x + \frac{d}{d x} [1]))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.22

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) (8 x + 0))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.23

Add $8 x$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + (2 x + 1) \cdot 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.24

Move $8$ to the left of $2 x + 1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 (4 x^{2} + 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + 8 \cdot (2 x + 1) x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25

Simplify.

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Step 12.3.25.1

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(2 \cdot 4 x^{2} + 2 \cdot 1) (2 x + 1) + (2 x - 1) (2 (4 x^{2} + 1) + 8 (2 x + 1) x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.2

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(2 \cdot 4 x^{2} + 2 \cdot 1) (2 x + 1) + (2 x - 1) (2 \cdot 4 x^{2} + 2 \cdot 1 + 8 (2 x + 1) x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.3

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(2 \cdot 4 x^{2} + 2 \cdot 1) (2 x + 1) + (2 x - 1) (2 \cdot 4 x^{2} + 2 \cdot 1 + (8 \cdot 2 x + 8 \cdot 1) x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.4

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(2 \cdot 4 x^{2} + 2 \cdot 1) (2 x + 1) + (2 x - 1) (2 \cdot 4 x^{2} + 2 \cdot 1 + 8 \cdot 2 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.5

Multiply $4$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2 \cdot 1) (2 x + 1) + (2 x - 1) (2 \cdot 4 x^{2} + 2 \cdot 1 + 8 \cdot 2 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.6

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (2 \cdot 4 x^{2} + 2 \cdot 1 + 8 \cdot 2 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.7

Multiply $4$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 \cdot 1 + 8 \cdot 2 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.8

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 8 \cdot 2 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.9

Multiply $2$ by $8$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x \cdot x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.10

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x^{1} x + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.11

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x^{1} x^{1} + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.12

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x^{1 + 1} + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.13

Add $1$ and $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x^{2} + 8 \cdot 1 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.14

Multiply $8$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (8 x^{2} + 2 + 16 x^{2} + 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.15

Add $8 x^{2}$ and $16 x^{2}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.16

Factor $1$ out of $(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x)$ .

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Step 12.3.25.16.1

Factor $1$ out of $(8 x^{2} + 2) (2 x + 1)$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 (8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.16.2

Factor $1$ out of $(2 x - 1) (24 x^{2} + 2 + 8 x)$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 (8 x^{2} + 2) (2 x + 1) + 1 (2 x - 1) (24 x^{2} + 2 + 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.16.3

Factor $1$ out of $1 ((8 x^{2} + 2) (2 x + 1)) + 1 ((2 x - 1) (24 x^{2} + 2 + 8 x))$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{1 ((8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x))}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.17

Multiply $(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x)$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(8 x^{2} + 2) (2 x + 1) + (2 x - 1) (24 x^{2} + 2 + 8 x)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.18

Reorder terms.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{(2 x + 1) (8 x^{2} + 2) + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19

Simplify each term.

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Step 12.3.25.19.1

Expand $(2 x + 1) (8 x^{2} + 2)$ using the FOIL Method.

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Step 12.3.25.19.1.1

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 x (8 x^{2} + 2) + 1 (8 x^{2} + 2) + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.1.2

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 x \cdot 8 x^{2} + 2 x \cdot 2 + 1 (8 x^{2} + 2) + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.1.3

Apply the distributive property.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 x \cdot 8 x^{2} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2

Simplify each term.

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Step 12.3.25.19.2.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 8 x \cdot x^{2} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.2

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

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Step 12.3.25.19.2.2.1

Move $x^{2}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 8 x^{2} x + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.2.2

Multiply $x^{2}$ by $x$ .

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Step 12.3.25.19.2.2.2.1

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 8 x^{2} x^{1} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.2.2.2

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 8 x^{2 + 1} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.2.3

Add $2$ and $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{2 \cdot 8 x^{3} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.3

Multiply $2$ by $8$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 2 x \cdot 2 + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.4

Multiply $2$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 1 \cdot 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.5

Multiply $8 x^{2}$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 1 \cdot 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.2.6

Multiply $2$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + (24 x^{2} + 2 + 8 x) (2 x - 1)}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.3

Expand $(24 x^{2} + 2 + 8 x) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 x^{2} \cdot 2 x + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4

Simplify each term.

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Step 12.3.25.19.4.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 \cdot 2 x^{2} x + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.2

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

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Step 12.3.25.19.4.2.1

Move $x$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 \cdot 2 x \cdot x^{2} + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.2.2

Multiply $x$ by $x^{2}$ .

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Step 12.3.25.19.4.2.2.1

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 \cdot 2 x^{1} x^{2} + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.2.2.2

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

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 \cdot 2 x^{1 + 2} + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.2.3

Add $1$ and $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 24 \cdot 2 x^{3} + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.3

Multiply $24$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} + 24 x^{2} \cdot - 1 + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.4

Multiply $- 1$ by $24$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 2 \cdot 2 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.5

Multiply $2$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x + 2 \cdot - 1 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.6

Multiply $2$ by $- 1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x - 2 + 8 x \cdot 2 x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.7

Rewrite using the commutative property of multiplication.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x - 2 + 8 \cdot 2 x \cdot x + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.8

Multiply $x$ by $x$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x - 2 + 8 \cdot 2 x^{2} + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.9

Multiply $8$ by $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x - 2 + 16 x^{2} + 8 x \cdot - 1}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.4.10

Multiply $- 1$ by $8$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 24 x^{2} + 4 x - 2 + 16 x^{2} - 8 x}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.5

Add $- 24 x^{2}$ and $16 x^{2}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 8 x^{2} + 4 x - 2 - 8 x}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.19.6

Subtract $8 x$ from $4 x$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 8 x^{2} - 4 x - 2}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20

Combine the opposite terms in $16 x^{3} + 4 x + 8 x^{2} + 2 + 48 x^{3} - 8 x^{2} - 4 x - 2$ .

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Step 12.3.25.20.1

Subtract $8 x^{2}$ from $8 x^{2}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 2 + 48 x^{3} + 0 - 4 x - 2}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20.2

Add $16 x^{3} + 4 x + 2 + 48 x^{3}$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 4 x + 2 + 48 x^{3} - 4 x - 2}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20.3

Subtract $4 x$ from $4 x$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 2 + 48 x^{3} + 0 - 2}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20.4

Add $16 x^{3} + 2 + 48 x^{3}$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 2 + 48 x^{3} - 2}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20.5

Subtract $2$ from $2$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 48 x^{3} + 0}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.20.6

Add $16 x^{3} + 48 x^{3}$ and $0$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3} + 48 x^{3}}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.25.21

Add $16 x^{3}$ and $48 x^{3}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{64 x^{3}}{\frac{d}{d x} [x^{4}]}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.3.26

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{64 x^{3}}{4 x^{3}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4

Reduce.

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Step 12.4.1

Cancel the common factor of $64$ and $4$ .

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Step 12.4.1.1

Factor $4$ out of $64 x^{3}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{4 \cdot 16 x^{3}}{4 x^{3}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4.1.2

Cancel the common factors.

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Step 12.4.1.2.1

Factor $4$ out of $4 x^{3}$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{4 \cdot 16 x^{3}}{4 (x^{3})}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4.1.2.2

Cancel the common factor.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{4 \cdot 16 x^{3}}{4 x^{3}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4.1.2.3

Rewrite the expression.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3}}{x^{3}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4.2

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

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Step 12.4.2.1

Cancel the common factor.

$\frac{\frac{- \sqrt{lim_{x \to - \infty} \frac{16 x^{3}}{x^{3}}}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 12.4.2.2

Divide $16$ by $1$ .

$\frac{\frac{- \sqrt{lim_{x \to - \infty} 16}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 13

Evaluate the limit.

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Step 13.1

Evaluate the limit of $16$ which is constant as $x$ approaches $- \infty$ .

$\frac{\frac{- \sqrt{16}}{lim_{x \to - \infty} 1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 13.2

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

$\frac{\frac{- \sqrt{16}}{1}}{lim_{x \to - \infty} 6 + \frac{1}{x}}$

Step 13.3

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

$\frac{\frac{- \sqrt{16}}{1}}{lim_{x \to - \infty} 6 + lim_{x \to - \infty} \frac{1}{x}}$

Step 13.4

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

$\frac{\frac{- \sqrt{16}}{1}}{6 + lim_{x \to - \infty} \frac{1}{x}}$

Step 14

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

$\frac{\frac{- \sqrt{16}}{1}}{6 + 0}$

Step 15

Simplify the answer.

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Step 15.1

Divide $- \sqrt{16}$ by $1$ .

$\frac{- \sqrt{16}}{6 + 0}$

Step 15.2

Simplify the numerator.

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Step 15.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{- \sqrt{4^{2}}}{6 + 0}$

Step 15.2.2

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

$\frac{- 1 \cdot 4}{6 + 0}$

Step 15.3

Add $6$ and $0$ .

$\frac{- 1 \cdot 4}{6}$

Step 15.4

Multiply $- 1$ by $4$ .

$\frac{- 4}{6}$

Step 15.5

Cancel the common factor of $- 4$ and $6$ .

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Step 15.5.1

Factor $2$ out of $- 4$ .

$\frac{2 (- 2)}{6}$

Step 15.5.2

Cancel the common factors.

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Step 15.5.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot - 2}{2 \cdot 3}$

Step 15.5.2.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 \cdot 3}$

Step 15.5.2.3

Rewrite the expression.

$\frac{- 2}{3}$

Step 15.6

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$