Answer to Question #115847 in Calculus for Ömür El

Question

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
a) A rational function has at most one vertical asymptote.
b) A rational function has at most one horizontal asymptote.
c) The graph of a rational function cannot cross a horizontal asymptote.

Answer 1

A rational function has at most one horizontal asymptote or an oblique (oblique) asymptote, and possibly many vertical asymptotes.

Vertical asymptotes occur only when the denominator is zero (If both the numerator and the denominator are zero, the multiplicity of zero is compared).

Therefore, the answer to part a is no because $\frac{2}{(x-2)(x-3)}$ has two verical asymptotes, that is, more than one vertical asymptote. The answer to part b is yes. The answer to part с is no since for example, $f(x)=\frac{2x^3 +5x^2 +1}{x^2 +1}$ has an asymptote $y=2x+5$ that crosses the graph.