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Numerade Educator

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Problem 62 Hard Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} x^{(\ln 2)/(1 + \ln x)} $

Answer

2

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Video Transcript

We're trying to find the limit as X approaches infinity of the function. And as normal, we are going to want to take the natural log of both sides to put it in a simpler form. That simpler form is going to be the natural log of two times the natural log of X divided by one plus the natural log of X. Then when we evaluate this at infinity we get infinity over infinity. This is an indeterminate form. So we're going to want to use low tells rule when we do this in the numerator will get Natural log of two times one over X. And then in the denominator we will just get one over X. So if that one over acts in one of our X will cancel out giving us just natural log of two as a result. Um So now that we know that's natural log of two, we also know that the natural log of why? Because we had to take the natural log of both sides. So we have that the natural log of Y equals the natural log of two. Based on that, it's clear that why is going to be equal to two? So chew is our final answer.