Determine the following limits.
lim w→∞ (ln w²

Question

Determine the following limits.

lim w→∞ (ln w²) / (ln w³ + 1)

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as x approaches infinity of LN of X to the power of 5th divided by LN of X to the power of 7th plus 4. And we are given 4 answer choices A says 1 divided by 4, B 5 divided by 7, C 0, and D 5 divided by 11. So let's consider this limit. We are given limit as x approaches infinity of LN of X to the power of 5th divided by LN of X to the power of 7th plus 4. And if we apply the direct substitution while x approaches infinity, so n of infinity approaches infinity, and we get the same in the denominator. So we get infinity divided by infinity, which is an indeterminate form. So let's simplify the function and let's apply the rules of logarithms. And specifically, we're going to apply the power rule. We're going to get limit as X approaches infinity, and according to the power rule, well, we can bring that 5 and that 7 in front of the log. This becomes a coefficient, right? So now we can rewrite the functions 5 multiplied by LN of X. Divided by 7 LN of X + 4. Now if we substitute infinity once again, well, nothing really changes, we get infinity divided by infinity. This is still an indeterminate form. So what we're going to do from here is divide all of our terms by LNX. We're going to get limit as x approaches infinity of 5 divided by 7 + 4 divided by LN X. We have divided all of our terms by LN of X, and now once again we can apply the direct substitution. We're going to get 5 divided by 7 plus 4 divided by. Now if x approaches infinity, then alan of x approaches infinity as well, so we get 4 divided by infinity. And 4 divided by infinity while this term approaches 0. We are dividing a number by an infinitely large number. So we end up with our final answer, which is 5 divided by 7, and the correct answer choice to this problem corresponds to the answer choice B. Thank you for watching.

Determine the following limits.
lim w→∞ (ln w²) / (ln w³ + 1)

Key Concepts

Limits

Natural Logarithm

L'Hôpital's Rule

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