Question

What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?

Answer 1

A rational function is where there are `x`'s under the fraction bar.

The part under the bar is called the denominator .
This puts limits on the domain of `x`, as the denominator may not work out to be `0`

Simple example: `y=1/x` domain : `x!=0`
This also defines the vertical asymptote `x=0`, because you can make `x` as close to `0` as you want, but never reach it.

It makes a difference whether you move toward the `0` from the positive side of from the negative (see graph).

We say `lim_(x->0^+) y=oo` and `lim_(x->0^-) y=-oo`

So there is a discontinuity
graph{1/x [-16.02, 16.01, -8.01, 8.01]}
On the other hand: If we make `x` larger and larger then `y` will get smaller and smaller, but never reach `0`. This is the horizontal asymptote `y=0`

We say `lim_(x->+oo) y=0` and `lim_(x->-oo) y=0`

Of course ratinal functions are usually more complicated, like:
`y=(2x-5)/(x+4)` or `y=x^2/(x^2-1)` but the idea is the same

In the latter example there are even two vertical asymptotes, as

`x^2-1=(x-1)(x+1)->x!=+1 and x!=-1`
graph{x^2/(x^2-1) [-22.8, 22.81, -11.4, 11.42]}