Question

 

In your own words, describe why the graph of a rational function can cross its horizontal asymptote but it can never cross a vertical asymptote.

Answer 1

It follows from the fact that you are graphing a function.

A function can only have a single value for any given input.

When we say an x value is not in the domain (because it is the location of a discontinuity such as a vertical asymptote or hole), we mean that the function is not defined there.

Meanwhile, if we say a curve “crosses” a given line (such as a vertical line of the form x=c), we mean there is a point on it that satisfies that line's equation (which in this case means the function is defined at x=c).

As you can see, “there is a vertical asymptote at x=c" is a direct contradiction to the claim “the function crosses the line x=c".

Horizontal asymptotes are fundamentally different because they do not impinge on the inputs to the function (and the inputs/domain are the only things restricted between the definition of a function and a relation). A function is allowed to have the same y value as many times as it wants or not at all. How it behaves as inputs become infinitely large or small has no influence on how it behaves in between because nothing in the definition of a function says “It can either have this y value once or not at all.” Indeed, a function is allowed to cross its horizontal asymptote(s) infinitely often.

Now, on the other hand, if you want to add such a restriction, you can. Defining a bijection requires you to impose exactly this restriction on y values, and therefore, a (mostly) continuous bijection will never cross its horizontal asymptote(s).

If this made sense at all, you should be able to come up with a satisfactory explanation as to why a function can have infinitely many vertical asymptotes but never more than two horizontal asymptotes.