Cubic asymptotes in rational functions?

[jann95]

I just got this assignment for math and the question was is it possible to have a cubic asymptote in a rational function. If so explain how and where.

 

[LCKurtz]

I just got this assignment for math and the question was is it possible to have a cubic asymptote in a rational function. If so explain how and where.

What is a "cubic" asymptote?

 

[jann95]

im not really sure, but we have to figure it out. Its like vertical, horizontal and oblique asymptotes in rational function. I have to figure out if there could be a cubic asymptote. I know that there could be a quadratic asymptote, but I am not sure about cubic.

 

[LCKurtz]

Then, what is a "quadratic" asymptote? Those terms is unfamiliar to me regarding asymptotes.

 

[jann95]

A quadratic asymptote is an asymptote on a rational function which looks like a parabola. So if you are graphing a rational function with a quadratic asymptote is doesn't pass through. Just like the oblique, vertical and horizontal asymptotes.

 

[LCKurtz]

A quadratic asymptote is an asymptote on a rational function which looks like a parabola. So if you are graphing a rational function with a quadratic asymptote is doesn't pass through. Just like the oblique, vertical and horizontal asymptotes.

Asymptotes don't "look like parabolas". They are straight lines. And I am certain your text doesn't define a quadratic asymptote as one that "looks like a parabola". Tell me, word for word, what your text definition of a quadratic asymptote is. How can you hope to solve a problem if you don't know the definitions?

 

[jann95]

For example 2x^3/x-1, when you divide 2x^3 by x-1 you will get a quadratic function, when you graph that function, that will be the quadratic asymptote.

 

[amgeorg]

I found an example of a cubic asymptote on http://www.webgraphing.com/graphing_basic.jsp
The rational function was f(x)= (2x^4)/(x-1).
Now I am curious if you can have asymptotes of any degree...

 

[Mark44]

I haven't run across this term (cubic asymptote) before, but I would guess that it means that for large |x|, the graph approaches that of some cubic polynomial.

 

[LCKurtz]

I haven't run across this term (cubic asymptote) before, but I would guess that it means that for large |x|, the graph approaches that of some cubic polynomial.

Yes, that is the obvious guess, and similarly for a "quadratic" asymptote. But it annoys me when posters won't look up definitions in their own texts and instead abandon the thread as jann95 has apparently done.

 

[Scollier]

Yes, you can have any degree asymptote. If the degree of the numerator is 3 greater than the degree of the denominator, you can have a cubic asymptote. These nonlinear asymptotes are called curvilinear asymptotes.

https://en.wikipedia.org/wiki/Asymptote#cite_note-9
http://www.wolframalpha.com/input/?i=curvilinear+asymptote

 

[HallsofIvy]

Asymptotes don't "look like parabolas". They are straight lines. And I am certain your text doesn't define a quadratic asymptote as one that "looks like a parabola". Tell me, word for word, what your text definition of a quadratic asymptote is. How can you hope to solve a problem if you don't know the definitions?

No, and asymptote does NOT have to be a straight line. Any curve can be an asymptote to a graph. The only requirement is that the graph, as x-> a, must get arbitrarily close to the curve without reaching it.

 

[LCKurtz]

No, and asymptote does NOT have to be a straight line. Any curve can be an asymptote to a graph. The only requirement is that the graph, as x-> a, must get arbitrarily close to the curve without reaching it.

To quote from Wikipedia which, as we all know, is the ultimate source of correctness in mathematics :

"More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although usually the term asymptote by itself is reserved for linear asymptotes."