60–81. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_x→∞ ln ((x +1) / (x-1))

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the limit, use Lopital's rule if necessary, and we're given the limit as X approaches infinity of the natural log of the quantity of X + 5 in quantity, divided by the quantity of X minus 5 in quantity. We're given 4 possible choices as our answers. For choice A, we have 0. for choice B, we have infinity, for choice C, we have minus infinity, and for choice D, we have D and E, which stands for does not exist. Now, we're asked to evaluate the limit. As x approaches infinity of the natural log of the quantity of X + 5 in quantity divided by the quantity of X minus 5. And so, the first thing we're going to do is actually rewrite the argument of the natural log function. So this is going to be equal to the limit. As x approaches infinity of the natural log of the quantity, and here in the numerator, I'll rewrite this as X minus 5 + 10, and that whole quantity is divided by the quantity of X minus 5. In quantity. So simplify this expression, this is going to be equal to the limit. As X approaches infinity of the natural log of the quantity of 1. Plus 10 divided by the quantity of X minus 5 in quantity. Now, when we evaluate this limit, we see that as X approaches infinity, The quantity of 10 divided by the quantity of X minus 5 in quantity is going to approach 0. Since we're going to have a finite number divided by a really large number, so that quantity is going to approach 0, as the number in the denominator gets much, much larger as it approaches infinity. So that means that when we evaluate this limit, it's going to be equal to the natural log of the quantity of 1 plus 0. So, then we'll have the natural log of 1, which is going to be equal to 0. So that is the answer that we're looking for for this problem, and if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim_x→∞ ln ((x +1) / (x-1))

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60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))