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?

e. x

Accepted Answer

Welcome back everyone. Determine whether the following statement is true or false. As x approaches infinity, F of X equals 2 X grows faster than G of X equals LN X. A says true, and B says false. So, for this problem, let's assume that this statement is true. If it is true, then if we calculate the limit as X approaches infinity. Of G of X divided by F of X, it must be equal to 0, right? And we want to check if the statement holds. The reason why the limit must be zero is because F of X grows faster than G of X. So when X approaches infinity, our denominator dominates. What we're going to do is check if this limit is indeed zero. So we're going to calculate the limit as X approaches infinity of LN of X divided by 2 X. And if we simply Substitute X approaches infinity, we're going to get infinity in the numerator and infinity in the denominator. So what we can do is apply Lato's rule. Let's differentiate the numerator and the denominator. In the numerator, the derivative of LN of X is 1 divided by X, and in the denominator, the derivative of 2 X is simply 2. So now we can calculate the limit as X approaches infinity of 1 divided by 2 X. In the numerator, we have a constant and our denominator approach is infinity. One divided by infinity gives us a limit of 0. So, indeed, the statement is true, and the answer is a. 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?e. 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?
e. x