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Numerade Educator

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Problem 54 Medium 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 0^+} \left(\frac{1}{x} - \frac{1}{\tan^{-1} x} \right) $

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Video Transcript

for this problem, we see that if we take the limit of our function are given, function is one over x minus one over the inverse tangent X. But we're going to rewrite this as the arc tangent of x minus X. Try to buy the X times the ark tangent of acts. So what we do is we found a common denominator and combine them. This allows us to take the limit more easily. We see that when we take the limit as X approaches zero from the right, we get 0/0. So that's going to be an indeterminate form. That's okay though, because now each knees low tells role which ends up giving us 1/1 plus X squared minus one. And then on the bottom we'll end up getting are tangent X plus acts over one plus X squared. Now combining everything and taking the limit as X goes to zero from the right, We end up getting on here -2 times zero. We take the derivative again, we'll get negative two times zero, and then we'll get another. Um We'll get 0-plus two. So as a result of that, our final answer is going to be zero.