Pre-Algebra Examples

$\frac{\sqrt{x^{5}}}{\sqrt{x^{12}}}$

Step 1

Find where the expression $\frac{| \frac{1}{x^{3}} | \sqrt{x}}{x}$ is undefined.

$x \leq 0$

Step 2

Since $\frac{| \frac{1}{x^{3}} | \sqrt{x}}{x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{| \frac{1}{x^{3}} | \sqrt{x}}{x}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 3

Evaluate $lim_{x \to \infty} \frac{| \frac{1}{x^{3}} | \sqrt{x}}{x}$ 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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{| \frac{1}{x^{3}} | x^{\frac{1}{2}}}{x}$

Step 3.1.2

Factor $x$ out of $| \frac{1}{x^{3}} | x^{\frac{1}{2}}$ .

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

Step 3.1.3

Cancel the common factors.

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Step 3.1.3.1

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

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

Step 3.1.3.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.3.3

Cancel the common factor.

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

Step 3.1.3.4

Rewrite the expression.

$lim_{x \to \infty} \frac{| \frac{1}{x^{3}} | x^{- \frac{1}{2}}}{1}$

Step 3.1.3.5

Divide $| \frac{1}{x^{3}} | x^{- \frac{1}{2}}$ by $1$ .

$lim_{x \to \infty} | \frac{1}{x^{3}} | x^{- \frac{1}{2}}$

Step 3.1.4

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

$lim_{x \to \infty} \frac{| \frac{1}{x^{3}} |}{x^{\frac{1}{2}}}$

Step 3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{| \frac{1}{x^{3}} |}{x^{\frac{1}{2}}}$ approaches $0$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

Step 5

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

Cannot Find Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7