Why can't a function have both a horizontal asymptore and an oblique asymptote?

Question

Why can't a function have both a horizontal asymptore and an oblique asymptote?

Answer 1

It can. For instance, the function f(x) = x^(sgn (x)) + 1/x has a horizontal asymptote (it is asymptotic to 0 in the negative x direction), a vertical asymptote (at x=0), and an oblique asymptote (it is asymptotic to y=x in the positive x direction).

Answer 2

This is not a formal explanation, but just a thought.

If the graph had both a horizontal and oblique asymptote, I think it would have to curve in such a way that it would not pass the vertical line test for it to be a function.