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?

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]}