Question 391550
A vertical asymptote occurs when there exists an x such that f(x) = p/0, where p is nonzero. For example, the function {{{f(x) = (x+5)/(x-3)}}} has a vertical asymptote at x = 3, since you obtain f(3) = 8/0, but the function {{{f(x) = ((x+3)(x+2))/(x+2)}}} does not have a vertical asymptote at -2, since the function is equivalent to {{{f(x) = x+3}}} except at -2 (where a "gap" in the function exists). Also, substituting x = -2, we get 0/0 which is indeterminate.



On the other hand, a horizontal asymptote occurs when a function converges to a limit as x goes to infinity or negative infinity. In limit notation, a function has a horizontal asymptote if {{{lim(x->infinity, f(x))}}} exists (or if the limit as x approaches negative infinity exists). There are many types of functions with horizontal asymptotes, and many different ways to find them.