82–89. Comparing growth rates Determine which of the two funct...

Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

eˣ and 3ˣ

Accepted Answer

Welcome back, everyone. In this problem, we want to compare the growth rates of the function Y equals 113 X and Y equals 9 X to tell which function grows faster. A says both functions grow at the same rate because their exponents are similar. B says 9 of X grows faster than 11 3 X because the base of 9 to the X is larger. C says 11 to 3X grows faster than Y E equals 9 X because the exponential term E3X has a larger growth rate compared to E2 XLN3. And he says both functions grow at the same rate because they are both exponential functions. Now, if we're going to tell which function grows faster, we can compare their growth rates using Lopita's rule. So we can evaluate the limit of their ratio as X approaches infinity. So first, let's set up our ratio, OK? So, let's consider Let's consider the limit as x approaches infinity of 11 to 3 X divided by 9 X. Now, before we continue, it would be good for us to consider if we could write this in a simpler way. That might make it easier for us to evaluate the limit. So could we write them in a common form? Well, Recall that 9 X is equal to 3 of 2 X based on what we know about the laws of exponents. We also know that if we have a term being raised to the poor a to the power of B, we could rewrite it as e to the power of B and A. So applying that idea, if we have 3 to the power of 2 X, then we could rewrite it as E to the power of 2 X multiplied by LN 3. So in that case, 9 X really could be written as E to the power of 2 X LN 3. If we are able to do that, then we can rewrite our limit, as the limit as X approaches infinity of 11 E to the 3 X. OK, divided by an E. To the power of 2 XLN 3 and again using what we know about powers, if we're dividing two powers with the same base, then we can rewrite the power in the numerator, but they are subtracting. In other words, dividing two bases with the same power means we're subtracting powers. That's an easier way for me to say it. So then we could rewrite this as E 3X minus 2 XLN3. Which we could take it a step further. And right, it says the limit as X approaches infinity of X, 11 E to the X multiplied by 3 minus 2 and then 3. Now this is pretty useful because 3 minus 2 and then 3s are constant, OK? And if we work it out. Then it's gonna be equal to. So right back everything here. E, it's, well, this expression is approximately 0.8028. So we could write this as E to the port of 0.8, 028 X. OK? And now this is pretty good because here we notice that as X approaches infinity or power or term ore term will also approach infinity. Therefore, the ratio grows without bound, OK? So as X approaches infinity, E to the power of 0.80 to 8 X approaches infinity, which means the ratio. Grows without bone. And no, since it's growing, OK, since it's approaching infinity, that means the term in the numerator 11 e to the power of 3 X must be growing faster than the term in the denominator. Ye equals 9 to the power of x. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

eˣ and 3ˣ

Key Concepts

Exponential Functions

Growth Rate Comparison

L'Hôpital's Rule

Watch next