How Do You Determine Horizontal Asymptotes in Functions?

[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?

 

[HallsofIvy]

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.

 

[M98Ranger]

To find horizontal asymptotes, you need to look at the behavior of the function as x approaches positive and negative infinity. If the function approaches a constant value as x gets larger or smaller, then that constant value is the horizontal asymptote.

There is a derivative test for finding horizontal asymptotes, called the L'Hopital's rule. This rule states that if the limit of a function as x approaches infinity or negative infinity is of the form 0/0 or infinity/infinity, then you can take the derivative of both the numerator and denominator and evaluate the limit again. If the new limit still results in 0/0 or infinity/infinity, you can repeat the process until you get a non-indeterminate form. The resulting value will be the horizontal asymptote.

In the given example, y=2xe^-x^5, as x approaches infinity, the exponential term dominates and the function approaches 0. Therefore, the horizontal asymptote is y=0. There are no other horizontal asymptotes for this function.