Polynomial division separates a rational function into its long-range trend and a shrinking remainder.
For x≠0 the identity is exact. As |x| grows, the remainder 1/x tends to zero, so the vertical distance to y=ax+b tends to zero. This is a slant asymptote when a≠0 and a horizontal asymptote when a=0. The experiment follows positive x.
An asymptote describes a limit; a graph may cross some asymptotes in other examples.