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Numerade Educator

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Problem 41 Easy Difficulty

Find the limit or show that it does not exist.

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

Answer

$$\infty$$

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Video Transcript

to evaluate this limit. Not that we can rewrite this into the limit as X approaches infinity of we can combine these using the property of natural log that is Ellen of a minus Ln of B. This is the same as Ellen of a over B. And so using this property we say Ln of one plus X squared minus Ln of one plus X is the same as Ln of one plus X squared over one plus X. And then using limit loss, this is the same as L N of the limit as X approaches infinity of one plus X squared Over one Plus X. And then from here we one up factor of the variable with the highest exponents for the numerator and the same way we will do it in our denominator. And so we get Ln of limits as X approaches infinity of x squared times one plus one over X squared over we have X times you have one plus one over X. And this is equal to L N of we have limits as X approaches infinity of can cancel it. And next here, so we have x times one plus one over X squared over one plus one over X. Now evaluating at infinity we have natural log of we have infinity times one plus one over infinity over one plus one over infinity and we know that One over infinity approaches to zero. And so from here we have Ln of Infinity over one which is the same as Ellen of Infinity and this is just infinity. And so we have shown that the value of the limit is infinity.