Skip over navigation

Calculus AB: Applications of the Derivative

Problems for "Vertical and Horizontal Asymptotes"

Vertical and Horizontal Asymptotes

Curve Sketching

Problem : Find any vertical asymptotes for f (x) = .

A candidate for a vertical asymptote is the place where the denominator goes to zero, which in this case is x = 3 . We must take limits to prove that this is an asymptote.


    = - ∞  
    = + ∞  

This means that x = - 3 is a vertical asymptote of f .

Problem : Find any vertical asymptotes for f (x) = .


  =  
    = (x + 2) = 5  
  =  
    = (x + 2) = 5  

This point is not an asymptote, but merely a point of discontinuity in the graph.

Problem : Find any horizontal asymptotes for f (x) = .

To find horizontal asymptotes, we take limits at infinity:


  =  
    = = 1  
  =  
    = = 1  

So, the line y = 1 is a horizontal asymptote of this function at both positive and negative infinity.

Problem : Find any horizontal asymptotes for f (x) = .


  =  
    = = + ∞  
  =  
    = = - ∞  

So, f does not have any horizontal asymptotes.

Follow Us