Vertical Asymptotes

A vertical asymptote occurs at x = c when the following are all true

  1) f (c) is undefined  
  2) f (x) = ∞ or - ∞  
  3) f (x) = ∞ or - ∞  

Taken together, #2 and #3 mean that f "grows without bound" as it approaches x = c. This happens most often with a rational function at a value of x that leads to a denominator of zero. For example, consider f (x) = . f (x) is undefined at x = - 1.

  1) f (x) is undefined at x = - 1  
  2)  = - ∞  
  3)  = + ∞  

Thus, x = - 1 is a vertical asymptote of f, graphed below:

Figure %: f (x) = has a vertical asymptote at x = - 1

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches, but never touches as x approaches negative or positive infinity. If f (x) = L or f (x) = L, then the line y = L is a horiztonal asymptote of the function f. For example, consider the function f (x) = . This function has a horizontal asymptote at y = 2 on both the left and the right ends of the graph:

Figure %: f (x) = . Has a horizontal asymptote at y = 2

Note that a function may cross its horizontal asymptote near the origin, but it cannot cross it as x approaches infinity.

Intuitively, we can see that y = 2 is a horizontal asymptote of f because as x approaches infinity, f (x) = behaves more and more like f (x) = , which is the same as f (x) = 2. Although f behaves more and more like this, it never actually becomes this function, so y = 2 is approached but not reached.