# How do you find the asymptotes of y=|x-1|/|x-2|?

Mar 30, 2018

$\textcolor{b l u e}{x = 2}$

$\textcolor{b l u e}{y = 1}$

#### Explanation:

By definition of absolute value:

$y = \frac{x - 1}{x - 2}$ and $y = \frac{- \left(x - 1\right)}{- \left(x - 2\right)}$

So these are both

$y = \frac{x - 1}{x - 2}$

This is undefined for $x = 2$, so,

$\textcolor{b l u e}{x = 2} \setminus \setminus \setminus \setminus \setminus \setminus$ is a vertical asymptote.

We now check end behaviour:

For limits to infinity we only need be concerned with the highest powers of $x$, so:

$\frac{x - 1}{x - 2} \to \frac{x}{x} = 1$

as $x \to \infty$, $\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \frac{x - 1}{x - 2} \to 1$

as $x \to - \infty$, $\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \frac{x - 1}{x - 2} \to 1$

So the line:

$\textcolor{b l u e}{y = 1} \setminus \setminus \setminus \setminus$ is a horizontal asymptote

GRAPH: