Calculus Examples

$\frac{x}{\sqrt{x^{2} - 4}}$

Step 1

Find where the expression $\frac{x}{\sqrt{x^{2} - 4}}$ is undefined.

$- 2 \leq x \leq 2$

Step 2

Since $\frac{x}{\sqrt{x^{2} - 4}}$ $\to$ $- \infty$ as $x$ $\to$ $- 2$ from the left and $\frac{x}{\sqrt{x^{2} - 4}}$ $\to$ $\infty$ as $x$ $\to$ $- 2$ from the right, then $x = - 2$ is a vertical asymptote.

$x = - 2$

Step 3

Since $\frac{x}{\sqrt{x^{2} - 4}}$ $\to$ $\infty$ as $x$ $\to$ $2$ from the left and $\frac{x}{\sqrt{x^{2} - 4}}$ $\to$ $\infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

Step 4

List all of the vertical asymptotes:

$x = - 2, 2$

Step 5

Evaluate $lim_{x \to \infty} \frac{x}{\sqrt{x^{2} - 4}}$ to find the horizontal asymptote.

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Step 5.1

Simplify.

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Step 5.1.1

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

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

Step 5.1.2

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 $b = 2$ .

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

Step 5.2

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

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

Step 5.3

Evaluate the limit.

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Step 5.3.1

Cancel the common factor of $x$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

Move the limit under the radical sign.

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

Step 5.4

Apply L'Hospital's rule.

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Step 5.4.1

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

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Step 5.4.1.1

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

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

Step 5.4.1.2

Evaluate the limit of the numerator.

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Step 5.4.1.2.1

Apply the distributive property.

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

Step 5.4.1.2.2

Apply the distributive property.

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

Step 5.4.1.2.3

Apply the distributive property.

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

Step 5.4.1.2.4

Reorder $x$ and $- 2$ .

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

Step 5.4.1.2.5

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

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

Step 5.4.1.2.6

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

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

Step 5.4.1.2.7

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

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

Step 5.4.1.2.8

Simplify by adding terms.

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Step 5.4.1.2.8.1

Add $1$ and $1$ .

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

Step 5.4.1.2.8.2

Multiply $2$ by $- 2$ .

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

Step 5.4.1.2.8.3

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

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

Step 5.4.1.2.8.4

Subtract $4$ from $0$ .

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

Step 5.4.1.2.9

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

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

Step 5.4.1.3

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

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

Step 5.4.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 5.4.2

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

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

Step 5.4.3

Find the derivative of the numerator and denominator.

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Step 5.4.3.1

Differentiate the numerator and denominator.

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

Step 5.4.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 2$ and $g (x) = x - 2$ .

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

Step 5.4.3.3

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

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

Step 5.4.3.4

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

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

Step 5.4.3.5

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

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

Step 5.4.3.6

Add $1$ and $0$ .

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

Step 5.4.3.7

Multiply $x + 2$ by $1$ .

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

Step 5.4.3.8

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

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

Step 5.4.3.9

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

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

Step 5.4.3.10

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

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

Step 5.4.3.11

Add $1$ and $0$ .

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

Step 5.4.3.12

Multiply $x - 2$ by $1$ .

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

Step 5.4.3.13

Add $x$ and $x$ .

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

Step 5.4.3.14

Subtract $2$ from $2$ .

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

Step 5.4.3.15

Add $2 x$ and $0$ .

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

Step 5.4.3.16

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

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

Step 5.4.4

Reduce.

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Step 5.4.4.1

Cancel the common factor of $2$ .

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Step 5.4.4.1.1

Cancel the common factor.

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

Step 5.4.4.1.2

Rewrite the expression.

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

Step 5.4.4.2

Cancel the common factor of $x$ .

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Step 5.4.4.2.1

Cancel the common factor.

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

Step 5.4.4.2.2

Rewrite the expression.

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

Step 5.5

Evaluate the limit.

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Step 5.5.1

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

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

Step 5.5.2

Simplify the answer.

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Step 5.5.2.1

Any root of $1$ is $1$ .

$\frac{1}{1}$

Step 5.5.2.2

Divide $1$ by $1$ .

$1$

Step 6

Evaluate $lim_{x \to - \infty} \frac{x}{\sqrt{x^{2} - 4}}$ to find the horizontal asymptote.

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Step 6.1

Simplify.

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Step 6.1.1

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

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

Step 6.1.2

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 $b = 2$ .

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

Step 6.2

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

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

Step 6.3

Evaluate the limit.

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Step 6.3.1

Cancel the common factor of $x$ .

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.3.4

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

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

Step 6.3.5

Move the limit under the radical sign.

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

Step 6.4

Apply L'Hospital's rule.

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Step 6.4.1

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

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Step 6.4.1.1

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

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

Step 6.4.1.2

Evaluate the limit of the numerator.

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Step 6.4.1.2.1

Apply the distributive property.

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

Step 6.4.1.2.2

Apply the distributive property.

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

Step 6.4.1.2.3

Apply the distributive property.

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

Step 6.4.1.2.4

Reorder $x$ and $- 2$ .

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

Step 6.4.1.2.5

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

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

Step 6.4.1.2.6

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

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

Step 6.4.1.2.7

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

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

Step 6.4.1.2.8

Simplify by adding terms.

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Step 6.4.1.2.8.1

Add $1$ and $1$ .

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

Step 6.4.1.2.8.2

Multiply $2$ by $- 2$ .

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

Step 6.4.1.2.8.3

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

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

Step 6.4.1.2.8.4

Subtract $4$ from $0$ .

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

Step 6.4.1.2.9

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

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

Step 6.4.1.3

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

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

Step 6.4.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 6.4.2

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

Step 6.4.3

Find the derivative of the numerator and denominator.

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Step 6.4.3.1

Differentiate the numerator and denominator.

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

Step 6.4.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 2$ and $g (x) = x - 2$ .

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

Step 6.4.3.3

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

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

Step 6.4.3.4

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

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

Step 6.4.3.5

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

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

Step 6.4.3.6

Add $1$ and $0$ .

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

Step 6.4.3.7

Multiply $x + 2$ by $1$ .

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

Step 6.4.3.8

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

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

Step 6.4.3.9

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

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

Step 6.4.3.10

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

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

Step 6.4.3.11

Add $1$ and $0$ .

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

Step 6.4.3.12

Multiply $x - 2$ by $1$ .

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

Step 6.4.3.13

Add $x$ and $x$ .

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

Step 6.4.3.14

Subtract $2$ from $2$ .

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

Step 6.4.3.15

Add $2 x$ and $0$ .

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

Step 6.4.3.16

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

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

Step 6.4.4

Reduce.

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Step 6.4.4.1

Cancel the common factor of $2$ .

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Step 6.4.4.1.1

Cancel the common factor.

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

Step 6.4.4.1.2

Rewrite the expression.

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

Step 6.4.4.2

Cancel the common factor of $x$ .

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Step 6.4.4.2.1

Cancel the common factor.

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

Step 6.4.4.2.2

Rewrite the expression.

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

Step 6.5

Evaluate the limit.

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Step 6.5.1

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

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

Step 6.5.2

Simplify the answer.

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Step 6.5.2.1

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

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Step 6.5.2.1.1

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

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

Step 6.5.2.1.2

Move the negative in front of the fraction.

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

Step 6.5.2.2

Any root of $1$ is $1$ .

$- \frac{1}{1}$

Step 6.5.2.3

Cancel the common factor of $1$ .

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Step 6.5.2.3.1

Cancel the common factor.

$- \frac{1}{1}$

Step 6.5.2.3.2

Rewrite the expression.

$- 1 \cdot 1$

Step 6.5.2.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7

List the horizontal asymptotes:

$y = 1, - 1$

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: $x = - 2, 2$

Horizontal Asymptotes: $y = 1, - 1$

Cannot Find Oblique Asymptotes

Step 10