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Problems for "Vertical and Horizontal Asymptotes"

Problems for "Vertical and Horizontal Asymptotes"

Problems for "Vertical and Horizontal Asymptotes"

Problems for "Vertical and Horizontal Asymptotes"

Problems for "Vertical and Horizontal Asymptotes"

Problems for "Vertical and Horizontal Asymptotes"

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.