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Numerade Educator

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Problem 56 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 1^+} [\ln(x^7 - 1) - \ln(x^5 - 1)] $

Answer

$\ln \left(\frac{7}{5}\right)$

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Video Transcript

So this problem, we're gonna be taking the limit as X goes to one from the right of the following function. Since it's a natural log function and it's using subtraction, we're going to turn it into a division by the The property that natural logs have. So the function now is going to be X to the 7th minus one um divided by actually faith except fifth, that's one. And keep mind this whole thing is the natural log of all that. But thankfully with limit, we can move the limit inside and kind of forget about the natural log for a second. Then when we do this we would get a determinant form. So we have to use local tells role. Once we simplify it further, we end up getting this down to 7/5. Yeah. Mhm. So the final answer, we can't forget about our natural log. So the final answer is going to be the natural log of 7/5.