Algebra Examples

$\frac{x}{\sqrt{x^{2} + 1}}$

Step 1

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

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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

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Step 3.1

Reduce.

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Cancel the common factors.

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Step 3.1.3.1

Multiply by $1$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.3.4

Divide $x {(x^{2} + 1)}^{- \frac{1}{2}}$ by $1$ .

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

Step 3.1.4

Move ${(x^{2} + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

Rewrite ${(x^{2} + 1)}^{\frac{1}{2}}$ as $\sqrt{x^{2} + 1}$ .

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

Step 3.3

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

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

Step 3.4

Evaluate the limit.

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Step 3.4.1

Cancel the common factor of $x$ .

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

Step 3.4.2

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

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Step 3.4.2.1

Cancel the common factor.

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

Step 3.4.2.2

Rewrite the expression.

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Move the limit under the radical sign.

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

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

Step 3.6

Simplify the answer.

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Step 3.6.1

Simplify the denominator.

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Step 3.6.1.1

Add $1$ and $0$ .

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

Step 3.6.1.2

Any root of $1$ is $1$ .

$\frac{1}{1}$

Step 3.6.2

Divide $1$ by $1$ .

$1$

Step 4

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

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Step 4.1

Reduce.

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factors.

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Step 4.1.3.1

Multiply by $1$ .

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

Step 4.1.3.2

Cancel the common factor.

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

Step 4.1.3.3

Rewrite the expression.

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

Step 4.1.3.4

Divide $x {(x^{2} + 1)}^{- \frac{1}{2}}$ by $1$ .

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

Step 4.1.4

Move ${(x^{2} + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2

Rewrite ${(x^{2} + 1)}^{\frac{1}{2}}$ as $\sqrt{x^{2} + 1}$ .

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

Step 4.3

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

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

Step 4.4

Evaluate the limit.

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Step 4.4.1

Cancel the common factor of $x$ .

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

Step 4.4.2

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

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Rewrite the expression.

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

Step 4.4.3

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

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

Step 4.4.4

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

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

Step 4.4.5

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

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

Step 4.4.6

Move the limit under the radical sign.

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

Step 4.4.7

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

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

Step 4.4.8

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

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

Step 4.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

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

Step 4.6

Simplify the answer.

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Step 4.6.1

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

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Step 4.6.1.1

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

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

Step 4.6.1.2

Move the negative in front of the fraction.

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

Step 4.6.2

Simplify the denominator.

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Step 4.6.2.1

Add $1$ and $0$ .

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

Step 4.6.2.2

Any root of $1$ is $1$ .

$- \frac{1}{1}$

Step 4.6.3

Cancel the common factor of $1$ .

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Step 4.6.3.1

Cancel the common factor.

$- \frac{1}{1}$

Step 4.6.3.2

Rewrite the expression.

$- 1 \cdot 1$

Step 4.6.4

Multiply $- 1$ by $1$ .

$- 1$

Step 5

List the horizontal asymptotes:

$y = 1, - 1$

Step 6

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

Cannot Find Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 1, - 1$

Cannot Find Oblique Asymptotes

Step 8