In the graph of a rational function, the line can sometimes cross the asymptote...when does this happen?

Question

How can I recognize when the line breifly crosses the asymptote given the function?

Answer 1

There is no good way to recognize such functions unless you use calculus to find local maxima and minima. For example, the function
f(x)=x/(x^2 -1)
has an asymptote at y=0 but crosses that asymptote when x=0.

Unlike what one of the other posters said, a rational function has to have the same horizontal asumptote as x->-infty as when x->+infty. An algebraic function can, however have two different horizontal asymptotes. For example,
f(x)=x/sqrt(1+x^2)
has asymptotes y=-1 as x->-infty and y=1 as x->+infty.

Answer 2

I forget the function, but there's one that starts at zero, goes up to a number, like say 3, and then slopes down and out, to an asymptote at something else, like 1.

Answer 3

It just can happen if it is a function defined in "parts" or "intervals"

Otherwise not, because an asymptote by definition is a line that a function can reach only in the infinite.