Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (x^2)/(x-2)^2-1/x`?

Answer 1

x=0
x=2
y=1
graph{(x^3-(x-2)^2)/((x-2)^2*x) [-45.1, 47.4, -22.3, 23.93]}

Explanation:

There are two types of asymptotes:
Firstly, those which are not in the domain:
that is x=2 and x=0
Secondly, that have a formula: y=kx+q
I do it like this (there may be a different way to do it)
`Lim_(xrarroo)f(x)=Lim_(xrarroo)(x^3-(x-2)^2)/((x-2)^2*x)`

In the type of limit where `xrarroo` and power functions you look only for the highest power so `y=Lim_(xrarroo)(x^3.....)/(x^3.....)=1`
The same goes for `xrarr-oo`