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Problem 82 Hard Difficulty

A metal cable has radius $ r $ and is covered by insulation so that the distance from the center of the cable to the exterior of the insulation is $ R $. The velocity $ v $ of an electrical impulse in the cable is
$$ v = -c \left( \frac{r}{R} \right)^2 \ln \left( \frac{r}{R} \right) $$
where $ c $ is a positive constant. Find the following limits and interpret your answers.
(a) $ \displaystyle \lim_{R\to r^+}v $ (b) $ \displaystyle \lim_{r\to 0^+}v $


b) 0


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Video Transcript

for this problem. We're letting our bigger approach. Little are um So what we have is the equals negative C. Times little are over bigger squared after a lot of are over. All right. Based on this, Since big guards approaching little are we'd end up getting A one right here and a one right here. Natural log of one is zero. So this whole thing is just going to be zero as a result. And there's not going to be an indeterminate form in the second portion. We'll get something slightly different in terms of solving for it. So here we're going to have the same formula only This time we have are approaching zero from the right. So we do get the indeterminate form. So as a result, we're gonna rewrite this to be the natural log of are over big are provided by one over R squared. And we pulled out see over big R squared. Since this gives us an indeterminate form where you're going to use Lotos rule and this allows us to get it too big. R little R squared over to where are is approaching zero from the right, so that would just be zero times. Anything will be zero. So the second answer is also zero.

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