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Problem 42 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \bigl[\ln (2 + x) - \ln (1 + x) \bigr] $

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Video Transcript

find the limit of Ln of two plus x minus Ln of one plus X. As X approaches infinity, we first rewrite this into the limit as X approaches infinity of L N of two plus x over one plus X. And here we use the property of natural log Ln of a minus Ln f b. This is equal to Ellen of a over B and then you sing limit loss for natural log functions. We can rewrite this further into Ln of the limit as extra purchase infinity of two plus X over one plus X. Now factoring out the very well with the highest exponent for the inside of L N. We have Ln limit as X approaches infinity of X times two over X plus one. This all over X times one over X plus one. And from here we have Ln of the limit as X approaches infinity of and here we can cancel out the X and we get two plus X plus one over one plus xbox one. and evaluating its infinity, we have L. N. of two over infinity plus one over one over Infinity Plus one. No constant over infinity. We'll always approach to zero and saw this to over infinity zero As well as one over infinity. And so we have Ln of 1/1. This is equal to Alan of one, that's Equal to zero. And so this is the value of the limits.