Question

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

Answer 1

The vertical asymptote is `x=0` and oblique asymptote
`y=x`

Explanation:

A line `x=a` is a vertical asymptote of a function f(x) if

`lim_(x->a)f(x)=+-oo`

A line `y = b` is a horizontal asymptote of a function f(x) if

`lim_(x->+-oo)=b`

An oblique asymptote for a function f(x) has the formula

`y=cx+d`

where

`c=lim_(x->oo)f(x)/x` and `d=lim_(x->oo)(f(x)-cx)`

Hence we have that

`f(x)->+-oo , x->0`

and `c=limf(x)/x=lim(x^2-4)/x^2=1`

`d=lim((x^2-4)/x-x)=0`

Hence `x=0` and `y=x`