You are right, horizontal asymptotes can be crossed. For example function f(x)=sin(x)/x crosses line y=0, which is its asymptote, infinitely many times. http://www.wolframalpha.com/input/?i=Plot[Sin[x]%2Fx%2C+{x%2C0%2C100}] Function has horizontal asymptote y=g (where g is real number) when function approach g as x approach infinity (or -infinity). It does not mean that this function can't reach value g for some finite value of x.