Expert: Paul Klarreich - 9/12/2007

Question
why can a graph not have any more than two vertical asymptotes on it?

Answer
Questioner:   MariAnne
Category:  Calculus
Private:  No
 
Subject:  horizontal asymptotes
Question:  why can a graph not have any more than two vertical asymptotes on it?

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Hi, MariAnne,

There is no reason why a graph can't have lots of V.A.'s on it.  Just look at the graph of y = tan x.

HOWEVER, I think you meant to write: "why can a graph not have any more than two HORIZONTAL asymptotes on it?"

Now this is quite a difficult question, and I think the answer again, is:

There is no reason at all, UNLESS you mean to define a horizontal asymptote in this way:

The horizontal line  y = a  is a Horizontal Asymptote in the positive direction if when  x -> +infinity, all the y-values are close to a.

Likewise, for a H.A. in the negative direction.

Now the answer goes along these lines:

Suppose a graph has two horizontal asymptotes in the positive direction, y = a AND y = b.  Well, if  y = a is a H.A. then when x -> +infinity, all the y-values must be close to  a.  In that case, they cannot also be close to b, so y = b cannot be a H.A. in the +-dir.

So there can only be one H.A. in the +-dir.  

Similarly, there can only be one H.A. in the negative direction, therefore no more than two H.A.'s in toto.  ['In toto' means in all, not inside Dorothy's little dog.]