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.

2ˣ and 4ˣ⸍²

Accepted Answer

Welcome back, everyone. In this problem, in comparing the growth rates of the functions Y equals 5 X and Y equals 252 of X, which function grows faster or are their growth rates comparable? A says 5X grows faster than 25 of X because its exponent is smaller. B says says 25 of X grows faster than 5 X because the base of 25 is larger than 5. C says both functions grow at the same rate because the ratio is 1, and the D says 5 X and 252 X grow at different rates because their exponents are different. Now, to compare their growth rates, we can use Lapita's rule, but we need to first express both functions in terms of a command base and then evaluate their ratio as X approaches infinity. OK? So let's consider the limit. As X approaches infinity of 5 to the pore of X divided by 25 to the pore of 1/2 of X. Now we could rewrite them in the same base here if we write 25 in terms of base 5. So that is gonna be 5 X divided by 5 rates to the power of 2 multiplied by 1/2 of X. Not this here too will cancer too. And this is gonna be the limit as X approaches infinity of 5 X divided by 55 to the of X divided by 5 to the power of X. But these two are the same thing. So that means we're finding the limit as X approaches infinity of 1, which would be equal to 1. Since the limit is 1, we can conclude that the two functions grow at the same rate as X approaches infinity, and that makes sense because like we figured out, both of these functions have the same value. 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.2ˣ and 4ˣ⸍²

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

2ˣ and 4ˣ⸍²