Horizontal Asymptotes

1. Jan 7, 2004

roy5995

How do i find horizontal asymptotes?

Is there a derivative test for that?

for example how do i find the asymptotes for y=2xe^-x^5 i know that there has to be a horizontal one, are there any others?

2. Jan 7, 2004

HallsofIvy

Staff Emeritus
You CAN use derivatives if you really want to: Find the derivative and see where it is approaching 0.

But you don't need to do that. Since a horizontal line extends to infinity, the only way a graph can have a horizontal asymptote is if the function approaches that value as x goes to + or - infinity.

In the particular case y=2xe-x^5, as x gets large, x^5 is much larger so you have e to a huge negative power. Even with x multiplying that, it will go to 0. y= 0 is a horizontal asympote. On the other hand if x goes to - infinity, for x a huge negative number, x^5 is a much larger negative number so -x^5 is a huge positive number. e-5^x is huge and -xe-x^5 is a huge negative number. here is no horizontal asymptote as x goes to negaitve infinity.

y= 0 is the only horizontal asymptote.

If you differentiate y, you get y'= 2e-x^5- 10x^5 sup[-x^5[/sup]. Since an exponential dominates any power of x, that will be close to 0 for large positive x, just as I said before.