Precalculus Examples

$y = \frac{2 x}{\sqrt{x^{2} - 1}}$

Step 1

Find where the expression $\frac{2 x}{\sqrt{x^{2} - 1}}$ is undefined.

$- 1 \leq x \leq 1$

Step 2

Since $\frac{2 x}{\sqrt{x^{2} - 1}}$ $\to$ $- \infty$ as $x$ $\to$ $- 1$ from the left and $\frac{2 x}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the right, then $x = - 1$ is a vertical asymptote.

$x = - 1$

Step 3

Since $\frac{2 x}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the left and $\frac{2 x}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

Step 4

List all of the vertical asymptotes:

$x = - 1, 1$

Step 5

Evaluate $lim_{x \to \infty} \frac{2 x}{\sqrt{x^{2} - 1}}$ to find the horizontal asymptote.

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Step 5.1

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

$2 lim_{x \to \infty} \frac{x}{\sqrt{x^{2} - 1}}$

Step 5.2

Simplify.

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Step 5.2.1

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

$2 lim_{x \to \infty} \frac{x}{\sqrt{x^{2} - 1^{2}}}$

Step 5.2.2

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

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

Step 5.3

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

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

Step 5.4

Evaluate the limit.

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Step 5.4.1

Cancel the common factor of $x$ .

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Move the limit under the radical sign.

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

Step 5.5

Apply L'Hospital's rule.

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Step 5.5.1

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

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Step 5.5.1.1

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

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

Step 5.5.1.2

Evaluate the limit of the numerator.

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Step 5.5.1.2.1

Apply the distributive property.

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

Step 5.5.1.2.2

Apply the distributive property.

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

Step 5.5.1.2.3

Apply the distributive property.

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

Step 5.5.1.2.4

Reorder $x$ and $- 1$ .

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

Step 5.5.1.2.5

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

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

Step 5.5.1.2.6

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

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

Step 5.5.1.2.7

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

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

Step 5.5.1.2.8

Simplify by adding terms.

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Step 5.5.1.2.8.1

Add $1$ and $1$ .

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

Step 5.5.1.2.8.2

Simplify.

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Step 5.5.1.2.8.2.1

Multiply $x$ by $1$ .

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

Step 5.5.1.2.8.2.2

Multiply $- 1$ by $1$ .

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

Step 5.5.1.2.8.3

Add $- 1 x$ and $x$ .

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

Step 5.5.1.2.8.4

Subtract $1$ from $0$ .

$2 \frac{1}{\sqrt{\frac{lim_{x \to \infty} x^{2} - 1}{lim_{x \to \infty} x^{2}}}}$

Step 5.5.1.2.9

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

$2 \frac{1}{\sqrt{\frac{\infty}{lim_{x \to \infty} x^{2}}}}$

Step 5.5.1.3

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

$2 \frac{1}{\sqrt{\frac{\infty}{\infty}}}$

Step 5.5.1.4

Infinity divided by infinity is undefined.

Undefined

$2 \frac{1}{\sqrt{\frac{\infty}{\infty}}}$

Step 5.5.2

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.

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

Step 5.5.3

Find the derivative of the numerator and denominator.

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Step 5.5.3.1

Differentiate the numerator and denominator.

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

Step 5.5.3.2

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

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

Step 5.5.3.3

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

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

Step 5.5.3.4

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

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

Step 5.5.3.5

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

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

Step 5.5.3.6

Add $1$ and $0$ .

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

Step 5.5.3.7

Multiply $x + 1$ by $1$ .

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

Step 5.5.3.8

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

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

Step 5.5.3.9

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

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

Step 5.5.3.10

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

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

Step 5.5.3.11

Add $1$ and $0$ .

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

Step 5.5.3.12

Multiply $x - 1$ by $1$ .

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

Step 5.5.3.13

Add $x$ and $x$ .

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

Step 5.5.3.14

Subtract $1$ from $1$ .

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

Step 5.5.3.15

Add $2 x$ and $0$ .

$2 \frac{1}{\sqrt{lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [x^{2}]}}}$

Step 5.5.3.16

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

$2 \frac{1}{\sqrt{lim_{x \to \infty} \frac{2 x}{2 x}}}$

Step 5.5.4

Reduce.

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Step 5.5.4.1

Cancel the common factor of $2$ .

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Step 5.5.4.1.1

Cancel the common factor.

$2 \frac{1}{\sqrt{lim_{x \to \infty} \frac{2 x}{2 x}}}$

Step 5.5.4.1.2

Rewrite the expression.

$2 \frac{1}{\sqrt{lim_{x \to \infty} \frac{x}{x}}}$

Step 5.5.4.2

Cancel the common factor of $x$ .

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Step 5.5.4.2.1

Cancel the common factor.

$2 \frac{1}{\sqrt{lim_{x \to \infty} \frac{x}{x}}}$

Step 5.5.4.2.2

Rewrite the expression.

$2 \frac{1}{\sqrt{lim_{x \to \infty} 1}}$

Step 5.6

Evaluate the limit.

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Step 5.6.1

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

$2 \frac{1}{\sqrt{1}}$

Step 5.6.2

Simplify the answer.

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Step 5.6.2.1

Any root of $1$ is $1$ .

$2 (\frac{1}{1})$

Step 5.6.2.2

Cancel the common factor of $1$ .

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Step 5.6.2.2.1

Cancel the common factor.

$2 (\frac{1}{1})$

Step 5.6.2.2.2

Rewrite the expression.

$2 \cdot 1$

Step 5.6.2.3

Multiply $2$ by $1$ .

$2$

Step 6

Evaluate $lim_{x \to - \infty} \frac{2 x}{\sqrt{x^{2} - 1}}$ to find the horizontal asymptote.

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Step 6.1

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

$2 lim_{x \to - \infty} \frac{x}{\sqrt{x^{2} - 1}}$

Step 6.2

Simplify.

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Step 6.2.1

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

$2 lim_{x \to - \infty} \frac{x}{\sqrt{x^{2} - 1^{2}}}$

Step 6.2.2

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

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

Step 6.3

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

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

Step 6.4

Evaluate the limit.

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Step 6.4.1

Cancel the common factor of $x$ .

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

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

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

Step 6.4.5

Move the limit under the radical sign.

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

Step 6.5

Apply L'Hospital's rule.

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Step 6.5.1

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

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Step 6.5.1.1

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

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

Step 6.5.1.2

Evaluate the limit of the numerator.

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Step 6.5.1.2.1

Apply the distributive property.

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

Step 6.5.1.2.2

Apply the distributive property.

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

Step 6.5.1.2.3

Apply the distributive property.

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

Step 6.5.1.2.4

Reorder $x$ and $- 1$ .

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

Step 6.5.1.2.5

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

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

Step 6.5.1.2.6

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

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

Step 6.5.1.2.7

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

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

Step 6.5.1.2.8

Simplify by adding terms.

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Step 6.5.1.2.8.1

Add $1$ and $1$ .

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

Step 6.5.1.2.8.2

Simplify.

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Step 6.5.1.2.8.2.1

Multiply $x$ by $1$ .

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

Step 6.5.1.2.8.2.2

Multiply $- 1$ by $1$ .

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

Step 6.5.1.2.8.3

Add $- 1 x$ and $x$ .

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

Step 6.5.1.2.8.4

Subtract $1$ from $0$ .

$2 \frac{1}{- \sqrt{\frac{lim_{x \to - \infty} x^{2} - 1}{lim_{x \to - \infty} x^{2}}}}$

Step 6.5.1.2.9

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

$2 \frac{1}{- \sqrt{\frac{\infty}{lim_{x \to - \infty} x^{2}}}}$

Step 6.5.1.3

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

$2 \frac{1}{- \sqrt{\frac{\infty}{\infty}}}$

Step 6.5.1.4

Infinity divided by infinity is undefined.

Undefined

$2 \frac{1}{- \sqrt{\frac{\infty}{\infty}}}$

Step 6.5.2

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

Step 6.5.3

Find the derivative of the numerator and denominator.

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Step 6.5.3.1

Differentiate the numerator and denominator.

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

Step 6.5.3.2

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

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

Step 6.5.3.3

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

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

Step 6.5.3.4

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

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

Step 6.5.3.5

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

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

Step 6.5.3.6

Add $1$ and $0$ .

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

Step 6.5.3.7

Multiply $x + 1$ by $1$ .

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

Step 6.5.3.8

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

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

Step 6.5.3.9

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

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

Step 6.5.3.10

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

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

Step 6.5.3.11

Add $1$ and $0$ .

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

Step 6.5.3.12

Multiply $x - 1$ by $1$ .

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

Step 6.5.3.13

Add $x$ and $x$ .

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

Step 6.5.3.14

Subtract $1$ from $1$ .

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

Step 6.5.3.15

Add $2 x$ and $0$ .

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{2 x}{\frac{d}{d x} [x^{2}]}}}$

Step 6.5.3.16

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

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{2 x}{2 x}}}$

Step 6.5.4

Reduce.

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Step 6.5.4.1

Cancel the common factor of $2$ .

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Step 6.5.4.1.1

Cancel the common factor.

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{2 x}{2 x}}}$

Step 6.5.4.1.2

Rewrite the expression.

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{x}{x}}}$

Step 6.5.4.2

Cancel the common factor of $x$ .

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Step 6.5.4.2.1

Cancel the common factor.

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} \frac{x}{x}}}$

Step 6.5.4.2.2

Rewrite the expression.

$2 \frac{1}{- \sqrt{lim_{x \to - \infty} 1}}$

Step 6.6

Evaluate the limit.

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Step 6.6.1

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

$2 \frac{1}{- \sqrt{1}}$

Step 6.6.2

Simplify the answer.

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Step 6.6.2.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.6.2.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$2 \frac{- 1 (- 1)}{- \sqrt{1}}$

Step 6.6.2.1.2

Move the negative in front of the fraction.

$2 (- \frac{1}{\sqrt{1}})$

Step 6.6.2.2

Any root of $1$ is $1$ .

$2 (- \frac{1}{1})$

Step 6.6.2.3

Cancel the common factor of $1$ .

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Step 6.6.2.3.1

Cancel the common factor.

$2 (- \frac{1}{1})$

Step 6.6.2.3.2

Rewrite the expression.

$2 (- 1 \cdot 1)$

Step 6.6.2.4

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.6.2.4.1

Multiply $- 1$ by $1$ .

$2 \cdot - 1$

Step 6.6.2.4.2

Multiply $2$ by $- 1$ .

$- 2$

Step 7

List the horizontal asymptotes:

$y = 2, - 2$

Step 8

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1, 1$

Horizontal Asymptotes: $y = 2, - 2$

Cannot Find Oblique Asymptotes

Step 10