What is the Explanation for a Horizontal Asymptote Always Being Above the Curve?

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Discussion Overview

The discussion revolves around the concept of horizontal asymptotes in calculus, specifically examining the function y = (x^2 - 1) / (x^2 + 1) and the claim that the curve is always above the horizontal asymptote y = 1. Participants are exploring the implications of this claim and questioning the accuracy of a textbook description.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant calculates the limit of the function as x approaches infinity and concludes that the horizontal asymptote is y = 1.
  • The same participant expresses confusion about the book's claim that the curve is always above the asymptote, suggesting that substituting values shows the function is actually below the asymptote.
  • Several participants agree that the textbook is incorrect in its description of the asymptotic behavior.
  • One participant humorously speculates that a printing error may have caused the confusion in the book's illustrations.

Areas of Agreement / Disagreement

Participants generally agree that the textbook's explanation is incorrect, but there is no consensus on the reasons behind this or the implications of the asymptotic behavior.

Contextual Notes

Participants do not provide specific details on the assumptions or definitions that may affect their interpretations of horizontal asymptotes.

singleton
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Well I'm going through an introductory calculus book and right now I'm on the section of horizontal asymptotes.

Currently I'm stubbed on this:
y = (x^2 - 1) / (x^2 + 1)

I take the limit of the function as x increases or decreases without bound and come up with y = 1 being the horizontal asymptote. No problem so far.

However, the book sketches it so that the curve is always above the asymptote for both + and - infinite.

The only problem is that if you substitute values for both sides, won't it ALWAYS be under the asymptote (since its a fraction less than one)? There will always be a value of 2 more on the denominator than the numerator.

So, am I right and the book wrong or what is what ;)
 
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Yeah, seems like the book is wrong.
 
singleton,

i am also convinced your book is wrong


regards
marlon
 
The book is wrong. What a strange book. Can you tell us what book it is?
 
e(ho0n3 said:
The book is wrong. What a strange book. Can you tell us what book it is?

Calculus (MCA-OAF) by the education ministry of Ontario ;)
 
A math book written by an education ministry? That explains it!

(Probably, the picture got flipped at the printers.)
 
HallsofIvy said:
A math book written by an education ministry? That explains it!

(Probably, the picture got flipped at the printers.)

It isn't just the picture ;) It describes it the same way heh.
 

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