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?

a. log_2(x²)

Accepted Answer

Welcome back, everyone. Determine whether the following statement is true or false. As x approaches infinity, F of X is equal to logarithm of base 3 of X cubed grows slower than G of X equals LN X. A says true, and B says false. For this problem, let's begin by assuming that this statement is true. Then if we calculate the limit as X approaches infinity. Of F of X divided by G of X. Then it must be equal to 0 because G of X is assumed to grow faster than F of X, and therefore our denominator must dominate, right? We want to check if this limit is indeed zero. If it is, the statement is true. So what we're going to do is begin by changing the base of the log of F of X. We have F of X equals log. Of Base 3 of X cubed. Let's remember how we change the base. Specifically, we want to take the base E. So we're going to take LM of X cubed and divided by LM. Of the original base, which is LN of 3, right? We can also simplify and show that this is 3 LN of X divided by LN of 3. Using the power rule of logarithms. So now, let's go ahead and set up our limit. We get limit as X approaches infinity. F of X is now 3 LN of X. Divided by a line of 3. And we are dividing by G of X which is LN of X, right? What we can do is perform division. And specifically, performing division, we can cross out a line of acts. And we can simplify the limit. We get limit as X approaches infinity of we divided by LN of 3. This is now a constant, right? The limit of a constant is equal to the constant itself, and this constant is not equal to 0. Because it is not equal to 0, this statement is false. So the answer is B. 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?a. log_2(x²)

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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?
a. log_2(x²)