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?

c. 1/√x

Accepted Answer

Welcome back, everyone. Determine whether the following statement is true or false. As x approaches infinity, F of X equals a 1 divided by a square root of 3 X, grows slower than G of X equals LN X. A says true, and B says false. Now, for this problem, we can use limits to prove if the statement is true or false. We're going to evaluate the limit as X approaches infinity. And what we're going to do is simply start by assuming that this statement is true. If it is true and F of X grows slower, then G of X is going to be in the denominator and F of X is going to be in the numerator. This limit must be equal to 0 because our denominator dominates over the numerator, meaning, G of X grows faster than F of X and therefore the limit is equal to 0. So we want to check if this limit is indeed zero. If it is, then the statement is true. Let's evaluate the limit as X approaches infinity of F of X is a 1 divided by square root of 3 X. And we're dividing by G of X, which is LN of X. Let's go ahead and divide. We get limit as X approaches infinity of 1. Divided by a square root of 3 X which is multiplied by LN of X. And now let's analyze the behavior. In the numerator we have one. In the denominator, square root of 3XS X approaches infinity is going to approach infinity, where it's taking square root of 3 multiplied by infinity, which is square root of infinity or basically infinity. Multiplied by a land of infinity which is infinity. So, overall, that's 1 divided by an infinitely large number. By definition, if we have a constant divided by an infinitely large number, the limit is 0. So yes, we got a limit of 0, meaning the original statement is true, and the answer is 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?c. 1/√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?
c. 1/√x