Asymptotes

[gerryrains]

In your definition of asymptotes you state that an asymptote of a relationship y = f(x) is a line that f(x) approaches but never reaches. You then precede to use the function y = f(x) = (4x+2) / (x^2 + 1).

You state that there is a horizontal asymptote since as x--> infinity, y --> 0 but never reaches it. Of course that is absolutely correct. However you then state that there is a vertical asymptote. The graph does not show that and I can't imagine what that asymptote might be.

Since all vertical asymptote must be of the form x = a, would you please tell me what the value of a is for the equation x = a? It certainly cannot be x = 0, since when x = 0, y = f(x) = 2. Clearly you are restricting the domain to the Real, as opposed to the Complex, numbers and I am quite bewildered.

Any clarification would be greatly appreciated.

Thank you,

Gerry Rains

 

[Deleted member 4993]

In your definition of asymptotes you state that an asymptote of a relationship y = f(x) is a line that f(x) approaches but never reaches. You then precede to use the function y = f(x) = (4x+2) / (x^2 + 1).

You state that there is a horizontal asymptote since as x--> infinity, y --> 0 but never reaches it. Of course that is absolutely correct. However you then state that there is a vertical asymptote. The graph does not show that and I can't imagine what that asymptote might be.

Since all vertical asymptote must be of the form x = a, would you please tell me what the value of a is for the equation x = a? It certainly cannot be x = 0, since when x = 0, y = f(x) = 2. Clearly you are restricting the domain to the Real, as opposed to the Complex, numbers and I am quite bewildered.

Any clarification would be greatly appreciated.

Thank you,

Gerry Rains

(4x+2)/(x^2+1) does not have a vertical asymptote.

 

[stapel]

In your definition of asymptotes you state that....

Who is the "you" that you're talking to...?

Eliz.

 

[Deleted member 4993]

I guess s/he is referring to the "subject" -> "Algebra" tab - but I did not find the example (however there is a discussion about horizontal asymptote with the said function). Then again I did not read very carefully.

 

[mmm4444bot]

... an asymptote of a relationship y = f(x) is a line that f(x) approaches but never reaches ...

I agree that the graph of a function approaches, but never reaches, a vertical asymptote.

The graph of a function may sometimes cross a horizontal asymptote once or many times as it continually moves closer (in a futile attempt) toward becoming that line.

In my sloppy sketch below, the horizontal asymptote has equation y=1.

A horizontal asymptote describes behavior of a fuction as its independent variable becomes very large in absolute value, when the behavior is that the function's values become closer to some constant value "over the long haul".

This differs from a vertical asymptote, where the function's values become closer to either positive or negative infinity as the independent variable becomes closer to some fixed point on the real number line.

Cheers,

~ Mark

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