Use limit methods to determine which of the two given function...

Question

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Accepted Answer

Welcome back everyone. In this problem, we want to identify which of the following two functions grows faster. X to the 16th and 2.0001 to the power of X. Use limit methods. A says it's X to the 16th, B 2.0001 to the X, and C says the two functions have comparable growth rates. Now what does it mean to use limit methods to figure out which function grows faster? Well, if we evaluate the limit of the ratio of the two functions as x approaches infinity. In other words, if we're trying to find the limit as X approaches infinity of X 16th divided by 2.0001 to the power of X. And if the limit is, well, yes, if we're trying to find a limit here, sorry, if the limit is 0, then our denominator 2.0001 to the X grows faster than our numerator X 16th. If the limit is infinity, then X 16th grows faster, and if the limit is positive and finite, the two functions have comparable growth rates. So let's try and see if we can figure these limits out, OK? Now, here, if we go ahead and substitute directly, we'll notice that both the numerator and denominator approach infinity as X approaches infinity. And in this case, this would leave our limit in an indeterminate form, OK? So because it's in an indeterminate form using direct substitution, we can try to use Lapita's rule to evaluate. What does that mean? Well, recall OK. That it basically says, if we're finding the limit as X approaches infinity of FX divided by G of X, OK? In other words, the limit of this quotient. OK. And it's in an indeterminate form. Then The limit, OK, we can find the limit as X approaches, well, here, it doesn't have to be infinity. It could be any value. So let's just say A here, OK. As x approaches A, then the limit as x approaches A of F of X divided by G of X is going to be equal to the limit as x approaches A of the derivative of F of X divided by the derivative of G of X. So if we can differentiate our numerator and denominator, we can apply it to Lopita's rule, and that can help us to solve. Now if we let our numerator F of X be equal to X 16th, then it's derivative. Is going to be equal to 16 X 15th. On the other hand, for a denominator, G of X equals 2.0001 to the power of X, then its derivative is going to be equal to 2.0001 to the X multiplied by the natural log of 2.0001, OK? So with these derivatives then by Lapita's rule we could rewrite our limit as the limit as x approaches infinity of 16 x 15th. Divided by 2.0001 to the X multiplied by the natural log of 2.0001. Now where do we go from here? Well, we can continue to apply Lopita's rule until the numerator becomes a constant or a 0, OK? And now repeating this process 16 times, the numerator will eventually become a constant while the denominator remains an exponential function multiplied by a constant. In other words, eventually it will become the limit as X approaches infinity of 16 factorial divided by 2.0001 to the X multiplied by the natural log of 2.0001. is to the point of 16, OK, which, when we evaluate that will be equal to 0. Since our limit equals 0, that means our denominator grows faster than our numerator. Thus, 2.0001 to the power of X is going to be our answer. B is correct. Thanks a lot for watching everyone. I hope this video helped.

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ

Key Concepts

Limits

Growth Rates

Asymptotic Behavior

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