Problem 10 Medium Difficulty

Given that $ \displaystyle \lim_{x \to \pi}\csc^2 x = \infty $, illustrate Definition 6 by finding values of $ \delta $ that correspond to (a) $ M = 500 $ and (b) $ M = 1000 $.

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Video Transcript

This is problem number ten of this tour Calculus eighth edition, section two point four. Given that the limit as X approaches pi, of course he can. Squared X is equal to Community Hill Astri definition six by finding values of Delta that corresponded to a M equals five hundred and B m equals one thousand. So for a reference, we're going to write definition six, which were given limits and sex and purchase A of a function he called to infinity here. All right. Zero is less than the quantity X minus a absolutely of the quantity west in the delta. And then the function is greater than I am some value and and to help us with this ah, problem. I have provided a a sketch of the co sign Sorry, cause he can score it function, which is our FX in this case. And we see that the coast seeking squared function and is this shape where every multiple pie beginning at zero zero pie to pie or zero negative pining to buy There is no defined a portion of the graph and we see that the ah there is an infinite limited each of these asam toots so high The limit definitely approaches positive. Infinity as X approaches pine. But we're trying to listrik definition six using these. No, these terms here Then we have discussed that we're going to find Delta values in order. Teo ones that are related to Emma's five hundred m is one thousand. So what we do is we use a graphene tool. We take this function cause he connects and we throw a Corsican Sorry, Costigan squared X. We put this in any graphing calculator in any graphing tool and we're going to do is we're going to try to find it a value for don't tell that satisfies this condition. So let's take the first part party Emma's five hundred. So there will be a point here much, much higher than one. But such that fine y equals one hundred. Well, across this ako seek support function. And if we take this here, for example, this point where across is because he can square function, Erm we can estimate on and use a trace function using using a tracing tool on a graphing calculator to determine exactly where this point lines. And if we do that, we should see that this ah, is it is at approximately three point, the X values approximately three point zero nine seven came. And what deserves to show us is that the difference between three point zero nine seven and pine is this distance here between the function at a height of five hundred, the function and the Assam tote, or eh X equals a and this defined there were daughter because we know that for the function being greater than M in party Emma's five hundred So among the function is greater than him. As you can see in this region pi plus this delta Hi minus two Stilton rate symmetric about X equals pine and the way we determine no time or one is this difference This Ah, this small difference between fine and exactly that we found at the height of nine hundred five hundred and we see that this is approximately point zero four. So that is our first told. Other corresponds with this height of five hundred essentially for any delta equal to or less than point zero four. In the end, he arranged there, we're guarantee that the function is greater than the value five hundred. If we further increase our scope of the function to an M equals one thousand for partying. We see that we reach our A next value. That is approximately three point one one. So notice that we've got even closer to pie, which guarantee that our doctor is going to be even smaller, not much smaller but smaller. And we see that we're gonna approximate value of point zero three. So this makes sound sensitive. We increased our restriction on the function f co sequence cortex that it must be greater than a thousand. We must confirmed Delta that is smaller because it is in this region, this higher region. So essentially within point zero three core les from pine, the function is restricted to this don't mean to this range of f greater than a thousand for part B. So we've found our two Delta bis that correspond to party and party on this. These are our final answers