Pre-Algebra Examples

$\sec (\arcsin (x - 1))$

Find where the expression $- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}$ is undefined.

$x \leq 0, x \geq 2$

Since $- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $- \frac{\sqrt{x (- x + 2)}}{x (x - 2)}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

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

$x = 2$

List all of the vertical asymptotes:

$x = 0, 2$

Since the limit does not exist, there are no horizontal asymptotes.

No Horizontal Asymptotes

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

Cannot Find Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0, 2$

No Horizontal Asymptotes

Cannot Find Oblique Asymptotes