Question

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

Answer 1

Horizontal asymptote: `y=1`
Vertical asymptote: `x=2`

Explanation:

Domain:
`x-2!=0`
`D(f)=RR-{2}`
That means there may be vertical asymptote. We have to find one-sided limit in the point 2

`Lim_(xrarr2^+)(x^2-1)/(x-2)^2=3/0^+=+oo`

`Lim_(xrarr2^-)(x^2-1)/(x-2)^2=3/0^(-)=-oo`

Definition says: If the limit in the point which is not included in the domain has a finite value then it's not an asymptote
That means: We didnt't find finite value so there is an asymptote.

Vertical asymptote: `x=2`

Horizontal asymptote: in `+oo`
You have to find a limit of f(x). If you find a finite value then it's an asymptote. Some look like this: `y=k`, this is asymptote that is parallel to x axis. There is also asymptote in the form: `y=kx+q` (it's line)

`Lim_(xrarroo)(x^2-1)/(x-2)^2=Lim_(xrarroo)(x^2-1)/(x^2-4x+2)=`

`Lim_(xrarroo)(cancel(x^2)(1-1/x^2))/(cancel(x^2)(1-4/x+2/x^2))=1/1=1`
`y=1`

Horizontal asymptote: in `-oo`
It's the same: `y=1`