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?

c. ln(√x)

Accepted Answer

Welcome back everyone. As X approaches infinity, which function, if any, grows faster. F of X equals L N X squad or G of X equals LN X. AS F of X. B says G of X. C says they grow at the same rate, and D says cannot be determined. For this problem, let's remember that if we want to compare the growth rate, we want to evaluate the limit as X approaches infinity of the ratio of two functions. Now, what we can do is simply take F of X in the numerator and G of X in the denominator. It doesn't matter which one we take, we have several possible outcomes that we're going to analyze based on the answer that we get. So, let's go ahead and substitute each function. We get limit as X approaches infinity of L and X squared divided by LN of X. As x approaches infinity, each function approaches infinity, so immediately we have an indeterminant form, but what we can do is simply apply the power rule of logarithms for the numerator, right? And we can bring down the exponents to get to LN of X instead of LN of X2, which essentially allows us to cancel out the two natural logarithms. And we get a limit of a constant, which is that constant itself. Whenever we get a constant that is not equal to 0, we can conclude that the two functions grow at the same rate, and because 2 is not equal to 0, we can say that the correct answer is option C. These two functions grow at the same rate because the limit is 2, which is not equal to 0. 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?c. ln(√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?
c. ln(√x)