Evaluate the following limits. Use l’Hôpital’s Rule when it is...

Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (log₂ x - log₃ x)

Accepted Answer

Welcome back, everyone. We want to evaluate the limit as X approaches infinity of log 3 of X minus log base 6 of X. A says it's a half, B says it's 1, C2, and D says it's infinity. Now, to evaluate the limit, OK, before we think about starting to use Lapita we will, let's rewrite our logs in the same base. That will make it easier for us to solve. Now recall, OK. That If we have a log. Logba A of B, we could rewrite this as logba C of B, OK? Logba C of B divided by logba C of A, OK, where C represents a common log. So this is for any positive number C. This will help us to rewrite our expression in the same log, and that makes it easier to solve like we said. So let's use the natural log as the base, OK? And it could be any log. If we use the natural log, then we could rewrite log base 3 of X as the natural log of x divided by the natural log of 3. And we could write log base 6 of X. As the natural log of x divided by the natural log of 6. Now if we replace this expression with our logs here in the limit, yeah. Then we're going to have the limit as X approaches infinity. Of LNX. Divided by LN3 minus LNX divided by LN6. We could take this a step further. All right, this is the limit as X approaches infinity of LNX multiplied by 1 divided by LN3 minus 1 divided by LN6. We can factor out the natural log of X here. Now, since the expression, the natural log of X grows without bond as X approaches infinity, and 1 divided by LN3 minus 1 divided by LN6 are constants, so the factor within the parenthesis here is a constant, then that would imply. That we could take a step further. Well, let me write it down here. We could take a step further and write this as. 1 divided by LN 3 minus 1 divided by LN6 multiplied by the limit as X approaches infinity of the natural lag of X. We know that that tends to X, so it's really 1 divided by LN3 minus 1 divided by LN6 multiplied by infinity. And we know that this will eventually be equal to infinity. So what does this tell us? Well, the limit as X approaches infinity of log 3 X minus logba 6 of X is equal to infinity. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (log₂ x - log₃ x)

Key Concepts

Limits

Logarithmic Functions

l'Hôpital's Rule

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