# How do you find all the asymptotes for function f(x) = (x+1) / (x+2)?

May 31, 2015

Let's start with the vertical , the value that is "forbidden" for $x$

Clearly that must be $x = - 2$ as this would make the denominator $= 0$ and that is not allowed.

The horizontal is when $x$ becomes very large (either negative or positive). You will see that in that case the $+ 1$ and $+ 2$ will have a smaller and smaller effect on the outcome. So the whole thing tends to resemble

$\frac{x}{x} \approx 1$ more and more, or: ${\lim}_{x \to \pm \infty} \frac{x + 1}{x + 2} = 1$

So $y = 1$ is the horizontal asymptote:
graph{(x+1)/(x+2) [-10, 10, -5, 5]}