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Numerade Educator



Problem 85 Hard Difficulty

(a) Use a graph to guess the value of the limit
$ \displaystyle \lim_{n \to \infty} \frac {n^5}{n!} $
(b) Use a graph of the sequence in part (a) to find the smallest values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.001 $ in Definition 2.




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Video Transcript

So for part A, let's use a graph to find this limit and to the fifth divided by in factorial. So let's go to our graph here. And you can see once you get to about two ten term in the sequence, it looks like they started conversion to zero quickly. So my guess here is that the limit is zero. Yeah, So for part B, we want the smallest value of end such that the following is true in this case. In general, you should write and minus the limit, but from party A we know that this equal zero So we're looking for the smallest value of capital and such that this implies a M wishes already positive is less than epsilon. So for the first case, we'LL do absalon equals zero point one and let's go ahead and find the value. And in the calculator so zero point one, you can see that the first time that happens the first time you're below point one is when n equals ten. However, the definition requires little and bigger than end. So we should really do and equal to nine. And if you take any little and bigger than this And nine, That means you're starting at ten. And that's what we want. Now we'LL do epsilon equals zero point zero zero one So we'LL go ahead and have to zoom in a little more here until we get a point zero zero one showing up. No, zoom in a little more And there we go. Point zero zero one we see it. Now let's go back a little bit to the graph and here we are. We see an eleven, we're still above point zero zero one. But if we go over to twelve, then we're below point zero zero one. So the first time we get below point zero zero one is that twelve? So we should go ahead and make this eleven and then any little and bigger than eleven. This is a problem to say and is bigger than her equals a twelve. This is why we're subtracting what? From the end. Wait. And that results part B