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Numerade Educator

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Problem 66 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} (2x - 1)^{\tan(\pi x/2)} $

Answer

$e^{2 / \pi}$

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Video Transcript

for this problem. We're going to be taking the limit as X goes to one of two, X minus one to the tangent kayaks over to. We're gonna want to take the natural log on both sides. So to make things a little simpler and when we do that, we're going to end up getting um the natural log of q minus X Divided by co tangent of high x over two. So now we can evaluate this at x equals one. Um but we'll get 0/0 the indeterminate form. So now we're going to use locales role. We're going to take the derivative of the numerator and we get one over to mine attacks. And then we're gonna take the derivative of asking me a negative actually. And we're going to take the derivative the denominator and we end up getting a negative coast sequence squared. Okay. Hi X over two times. Hi over two as a result of this. When we evaluate it we end up getting chew over pie. But remember that that is the natural log of Y. Yeah. Equalling two over pie. So our final answer is going to be E to the two over. Hi, for the final answer.