Question

How do you identify the horizontal asymptote of `f(x) = (3)/(5x)`?

Answer 1

Try making `x` larger and larger and see where that leads you:

As `x` gets larger (either positive or negative) `f(x)` gets smaller. You can get as close to `0` as you want, but never get there.
So `f(x)=0` is the horizontal asymptote .
Or in "the language"
`lim_(x->oo) 3/(5x)=0` and `lim_(x->-oo) 3/(5x)=0`

Btw: `x=0` is the vertical asymptote , as `x` may not be `0`
`lim_(x->0^+) 3/(5x)=oo` and `lim_(x->0^-) 3/(5x)=-oo`
graph{3/(5x) [-10, 10, -5, 5]}