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

In summary, the conversation discusses the topic of horizontal asymptotes in calculus and the confusion caused by the book's sketch. The individual has found a discrepancy between their calculations and the book's depiction, leading them to believe the book is wrong. However, the book is a math book written by the education ministry of Ontario, which may explain the mistake.
  • #1
singleton
121
0
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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  • #2
Yeah, seems like the book is wrong.
 
  • #3
singleton,

i am also convinced your book is wrong


regards
marlon
 
  • #4
The book is wrong. What a strange book. Can you tell us what book it is?
 
  • #5
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 ;)
 
  • #6
A math book written by an education ministry? That explains it!

(Probably, the picture got flipped at the printers.)
 
  • #7
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.
 

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

What is a horizontal asymptote?

A horizontal asymptote is a line that a graph approaches but never touches. It represents the limit of the graph as the input value approaches infinity or negative infinity.

How do you find a horizontal asymptote?

To find a horizontal asymptote, you need to look at the highest degree terms in the numerator and denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

What does it mean if a function has no horizontal asymptote?

If a function has no horizontal asymptote, it means that the graph of the function does not approach a horizontal line as the input value approaches infinity or negative infinity. This could indicate that the function is unbounded and continues to increase or decrease without approaching a specific value.

Can a function have more than one horizontal asymptote?

Yes, a function can have multiple horizontal asymptotes. This occurs when the degrees of the numerator and denominator are the same, but the leading coefficients are different. In this case, the horizontal asymptotes are the different ratios of the leading coefficients.

How does a horizontal asymptote affect the behavior of a graph?

A horizontal asymptote can affect the behavior of a graph by limiting the values that the graph can approach. For example, if the horizontal asymptote is y = 0, the graph will never have a y-value of 0. This can also affect the end behavior of the graph, as it will either approach or move away from the horizontal asymptote as the input value increases or decreases.

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