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.

x² ln x; x³

Accepted Answer

Welcome back, everyone. In this problem, we want to identify which of the following two functions grows faster using limit methods Xcubed LNX and X to the 4th. A says it's X cubed LNX, B X 4th, and C says the two functions have comparable growth rates. Now, to determine which function grows faster, we can use the limit comparison method, which means we'll examine the limit of the ratio of the two functions as X approaches infinity. So in other words, let's consider the limit. As x approaches infinity of X cubed LN X. Divided by X to the 4th, OK? Where if our limit evaluates to 0, that means the denominator grows faster than the function in the numerator. And if it evaluates to infinity, that means the function in the numerator grows faster than the function in the denominator. OK? Now, we could simplify our limit here, since both of them have X cubed in common to be equal to the limit as X approaches infinity of LNX divided by X. Now if we were to directly substitute infinity into our uh limit here, OK, then in our numerator it would be infinity, and in our denominator it would be infinity. Thus our limit would be in an indeterminate form. So we can't figure out our limit by direct substitution. Instead, we'll have to use Lapital's rule, OK? Now, what does Lapita's rule say? Recall what it basically tells us is that if we have a limit. As X approaches A alpha quotient, FX divided by G of X, OK. That results in an indeterminate form. In other words, it's indeterminate. Then We can find the limit as X approaches A of FX divided by G of X. By finding the limit as x approaches a of the derivative of FX divided by the derivative of Gaffi. In other words, we can differentiate FX, GX, then sulfa or a limit. So let's find the derivatives of our numerator and denominator in our function, and then we'll apply Lapita's rule. No, that means if we let Efa fix. B equal to X cubed LN X. And if we let Ga fix. B equal to X to the 4th, OK. Then the derivatives, well, you know what, we could even write it in our simplest form actually. So if we let F of X be equal to the natural log of X and G of X be equal to X, OK, then the derivative of our numerator would be equal to 1 divided by X. And for a numerator, it would be equal to one. In this case, that means then that by Lapita's rule, we're going to be finding the limit as X approaches infinity. OK, this case, that would be a of 1 divided by X divided by 1. OK. That is basically going to be equal to the limit as X approaches infinity of 1 divided by X. And we know that that limit is going to be equal to 0. In other words, the the larger or denominator gets here, the closer our number or our limit gets to 0. So since this limit evaluates to 0, that means the denominator, OK, the denominator in our expression grows faster than the function in the numerator. Grows faster than the numerator, OK? So if we apply that to our problem, thus, that means X the 4th grows faster. than Ellen, Xcued LNX. Therefore, B is the correct answer. 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.
x² ln x; x³

Key Concepts

Limits

Growth Rates

L'Hôpital's Rule

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