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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.
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