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

Question

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

g. ln(ln x)

Accepted Answer

Welcome back, everyone. Determine whether the following statement is true or false. As x approaches infinity, F of X equals LN of 3, LN of X grows slower than G of X equals LN of X. A says true, and B says false. For this problem, let's begin with an assumption. Assume that this statement is true. If this statement is true, Then GFX grows faster. Then F of X, and therefore the limit as X approaches infinity of F of X divided by G of X must be zero, because G of X dominates against F of X, right? And we want to check if this limit is indeed 0. So, we can substitute each function to get a limit as X approaches infinity. Of LN of 3 LN of eggs. Divided by LM of X. To solve for this limit, we're going to use the properties of logarithms. In the numerator, we have LN of a product, right? So we can rewrite it as LN of 3. Plus LN of LN of X. And we're dividing by LM of X. So now we can split this limit into two limits, limits, X, approaches infinity. Of EN of 3. Divided by LN of X plus. Limit as x approaches infinity. Of Ellen, of Ellen of X. Divided by LN of X. For the first limit, notice that we have a constant in the numerator and LNF X as X approaches infinity. Approach his infinity A constant divided by infinity. is going to give us a limit of 0. So the first limit is 0, and we want to focus on the second limit. A land of x as x approaches infinity is going to approach infinity and the land of infinity. Is going to be infinity. So, again, we get infinity divided by infinity, which is an indeterminate form, right? In this indeterminate form is suitable for Lapito's rule. Meaning what we can do is differentiate the numerator and the denominator. Let's take the derivative of LN of LN of X, which is a composite function. So, first of all, we're differentiating LN with respect to LN of X, which gives us One divided by Alan of X. And based on the chain rule, we're multiplying by the derivative of the inner function. Which is the derivative of LN of X, and that's one divided by X. In the denominator, the derivative of LN X is equal to 1 divided by X, and we can clearly see that we can now cancel out 1 divided by X in each part of the fraction. So we now get limit as X approaches infinity of 1 divided by LN of X. Which leaves us with a constant in the numerator, that's one. And a land of infinity is going to approach infinity. 1 divided by infinity is 0, so this limit is 0. Which is consistent with our assumption, meaning the statement is true. And the correct answer is option A. Thank you for watching.

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

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