### Video Transcript

Compare the growth rates of the two functions π of π₯ equals π to the power of π₯ and π of π₯ equals the natural log of π₯ using limits as π₯ approaches β.

We begin by recalling that we can use limits to compare the growth rate of two functions by using the following definition. Let π of π₯ and π of π₯ be positive for values of π₯ sufficiently large. We say that π of π₯ grows faster than π of π₯ as π₯ approaches β if the limit as π₯ approaches β of the quotient π of π₯ over π of π₯ is equal to β. Or equivalently, if the limit as π₯ approaches β of π of π₯ over π of π₯ is equal to zero.

We can also conversely say that, given these situations, π of π₯ grows slower than π of π₯ as π₯ approaches β. This can be denoted as shown. Now, alternatively, if the limit as π₯ approaches β of π of π₯ over π of π₯ is equal to some finite nonzero number, we say that π of π₯ and π of π₯ grow at the same rate as π₯ approaches β.

So, we see weβre going to need to find the limit as π₯ approaches β of π of π₯ over π of π₯. And we can define this in any order we like. But, of course, weβve been told that π of π₯ is equal to π to the power of π₯ and π of π₯ is equal to the natural logarithm of π₯. So, letβs have a look at the limit as π₯ approaches β of π to the power of π₯ over the natural logarithm of π₯.

Now, we notice that if we try direct substitution, we obtain β over β, which is of course of indeterminate form. We are alternatively then going to use LβHΓ΄pitalβs rule. The part of the rule that weβre interested in says that if the limit as π₯ approaches some π of π of π₯ over π of π₯ is equal to β over β, where π itself can be a real number β or negative β. Then the limit as π₯ approaches π of π of π₯ over π of π₯ is equal to the limit as π₯ approaches π of π prime of π₯ over π prime of π₯. And this is only true when π and π are differentiable around π.

So, this rule essentially tells us that if we have the indeterminate form β over β β and in fact, zero over zero β all we need to do is differentiate the numerator and denominator and then take that limit. So, we begin by differentiating π to the power of π₯. And of course, thatβs simply π to the power of π₯. Weβll also need to differentiate the denominator. Thatβs the natural logarithm of π₯. And when we differentiate that with respect to π₯, we get one over π₯. So, we need to work out the limit as π₯ approaches β of π to the power of π₯ over one over π₯.

Now, it might be sensible to write π to the power of π₯ over one over π₯ as π to the power of π₯ divided by one over π₯. And of course, we know that when we divide by a fraction, we multiply by the reciprocal of that same fraction. So, weβre actually looking for the limit as π₯ approaches β of π to the power of π₯ times π₯ over one, or the limit as π₯ approaches β of π₯ times π to the power of π₯.

Well, the limit as π₯ approaches β of this is itself β. We go back to our earlier definition. And we see that the limit as π₯ approaches β of π of π₯ over π of π₯, which weβve evaluated using LβHΓ΄pitalβs rule. Weβve seen that the limit as π₯ approaches β of π prime of π₯ over π prime of π₯ is β. This means our function π of π₯ grows faster than our function π of π₯. And we can, therefore, say that the growth rate of the function π to the power of π₯ is greater than the growth rate of the natural logarithm of π₯.