5. Which of the following functions grow faster than ln(x) as ...

Question

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_3(x)

Accepted Answer

Welcome back, everyone. As X approaches infinity, which function, if any, grows faster. F of X equals a log of base 2 of X or G of X equals LN X. As F of X B G of X, they grow at the same rate, and D cannot be determined. For this problem, let's remember that if we want to compare the growth rate of two functions, we have to take the limit. S X approaches infinity of their ratio. And what we're going to do is simply take the ratio of F divided by G. It doesn't matter which one is the numerator and which one is the denominator, we're going to analyze the outcome later. So We are going to get limit as X approaches infinity, F of X is log of base 2 of X. And G of X is LN X. As X approaches infinity, each logarithm approaches infinity, so we're going to get an indeterminant form infinity divided by infinity. And to avoid that, we can essentially change the base of a log based to of X. Let's turn it into the natural log. That would be LN of X divided by LN of 2. And now, if we divide by LN of X, we are going to get limit as X approaches infinity of LNX divided by LN of 2 divided by LN of X is going to be 1 divided by LN of 2, because we can cancel out LN of X in the numerator and LN of X in the denominator. And now 1 divided by Alan of 2 is a constant. The limit of a constant is equal to the constant itself. So, our limit is 1 divided by LN of 2. And whenever our limit is a constant, except from 0, we can conclude that the two functions grow at the same rate, because one divided by a line of 2 is not 0, the answer is option C. Thank you for watching.

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?a. log_3(x)

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5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)