Question

Question:

Why can't a rational function have both, a horizontal and an oblique asymptote?

Could someone give a non-wikipedia answer?

Thanks

Answer 1

@danglade Just wondering - why do you think a rational function cannot have both a horizontal and an oblique asymptote? Or, why not horizontal, vertical, and oblique asymptotes?

I agree. Wikipedia is a non-academic tool.

Answer 2

Thats the reason I'm asking, there is not a formal explanation. I'm looking for the most correct one

Answer 3

I would create a rational function and try to fashion it so that it has a horizontal and an oblique asymptote. I have not seen a formal expression that answers your question but there may be one.

Answer 4

Any example please?

Answer 5

I don't have one in my head.
I'm thinking but am not certain that rational functions with horizontal asymptotes have the same power of the variable in the numerator as the denominator. And, I'm also thinking but am not certain that rational functions with oblique asymptotes have a power of x in the numerator that exceeds the power of x in the denominator. Of course, there are other points to consider. Y = (x^2 - 1) / (x - 1) does not have an oblique asymptote. What I'm going to do is to play with factors and see if I can find a rational function like the one you described.

Answer 6

This is a great question and indicates that someone is thinking deeply about mathematics. So, count me in to probe the matter.

Answer 7

Thanks! I'll wait!

Answer 8

Okay, while you are waiting, play around with factors and build rational functions. See if you can get the an example of the function we seek.

Answer 9

Let say \[(6x ^{2} + 7x + 4)/(2x^2 - 3x - 5)\]