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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.$\lim _{x \rightarrow 1} \frac{1-x+\ln x}{1+\cos \pi x}$

$-\frac{1}{\pi^{2}}$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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All right, So for this problem, we are evaluating the limit as exit purchase of one of one minus X plus the natural log of X over one plus coast and avec. So we're going to start by evaluating the limit. And I will give us one minus one plus the natural log of one over one plus co sign pi times one. Okay. And that's gonna equal one minus one plus 0/1 minus one. Okay, which is gonna be 0/0? So loopy calls. Rule applies, and we can use it. And so that means that our original limit will be equivalent to the limit. As X approaches, one of the derivative of one will just fall off. But we'll get a negative one from the X, and they will take the derivative of the Alana Natural Argon X. That's me one over X. Move on to the denominator, and so one will turn into zero then, of course, the co sign of Pi X is gonna be negative. Sign of Pi X. Okay, so we've taken that. We're gonna evaluate the limit again. So we'll put one in, and we get a negative one plus one minus sign of high times One. Okay. And that's gonna be 0/0. So we can fly low petals, rule a second time, and so that gives us the limit. Has X of purchase one. The one take the evidence of that becomes a zero. The derivative of one over axe is gonna be negative. One over x squared, And then we take the derivative of the negative part, as it should be a negative pie time. Sinan X. Okay, so then when we take the derivative of the negative pi times Native Pyatt times a sign of Pi X that's going to give us negative pi squared Times co sign. Hi, ex. So let's evaluate the limit for 1/3 time. And so we put a one in here. We get just a negative one on the top. And on the bottom, we're gonna get a native pi squared. Hi, I'm c co sign of tie. I'm just gonna give us a negative one over negative pi squared at times. Negative one. Okay, so it ends up being negative one over hi squared. And so that's going to be our limit

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