Precalculus Examples

$f (x) = \sqrt{\frac{4 x + 1}{x - 4}}$

Step 1

Find where the expression $\sqrt{\frac{4 x + 1}{x - 4}}$ is undefined.

$- \frac{1}{4} < x \leq 4$

Step 2

Since $\sqrt{\frac{4 x + 1}{x - 4}}$ $\to$ $\infty$ as $x$ $\to$ $4$ from the left and $\sqrt{\frac{4 x + 1}{x - 4}}$ $\to$ $\infty$ as $x$ $\to$ $4$ from the right, then $x = 4$ is a vertical asymptote.

$x = 4$

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

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: $x = 4$

No Horizontal Asymptotes

Cannot Find Oblique Asymptotes

Step 7