Question

How do you find the asymptotes for `y = (x^2-2x)/(x^2-5x+4)`?

Answer 1

The asymptote is `y = 1`.

Explanation:

In order to know if `f(x) = (x^2 - 2x)/(x^2-5x+4)`, you need the limit of `f` when `x` becomes infinite.

We know that in a rational function, only the highest power matters at the infinites. So we can say that `lim_(+oo) f = 1` and it means that the line `y = 1` is an asymptote when `x` becomes really big, and it is the exact same thing when `x -> -oo`. It's quite visible on the graph.

graph{(x^2 -2x)/(x^2 - 5x + 4) [-8.66, 11.34, -3.24, 6.76]}