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(a) Estimate the value of $\quad \lim _{x \rightarrow 0} \frac{\sin x}{\sin \pi x}$ by graphing the function $$f(x)=(\sin x) /(\sin \pi x) .$$Stateyour answer correct to two decimal places.(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach $0 .$

(a) $\lim _{x \rightarrow 0} f(x) \approx 0.32$(b) $\approx 0.318310 \approx 0.32$

Calculus 1 / AB

Chapter 2

Limits

Section 3

Limits of Functions at Finite Numbers

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

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everybody we're doing Chapter two, Section three Problem 28. We are doing the limit as X approaches. Zero of sign of X over Sign Pi X. So if I just quickly, you know, draw it out, at least for you know what it looks like? The reason why this is such a problem because you are trying to find as X approaches zero. Then if you plug in zero, right, If we're talking about limits, we're tryingto find out trends as we get to zero. But if you're trying to think about like Okay, well, what is it? A zero? You plug in zero for this, you get sign of zero, which is a row. You get signed of pi times zero, which is also zero. So you get zero divided by zero. So if you divide something by zero, it's undefined. If you take zero divided by something, it's zero. So you get zero divided by something. It's undefined. You get an unknown quantity. It's this very weird thing that can happen. So that's why this ends up being a really interesting problem. The cool thing about it is, that part is is saying, Hey, graft this thing right, and then look at it. And so, if you if you pull out your favorite graphing, you know, device, whatever you happen to have, um you kind of zoom in on this thing, Then you'll end up finding that right here on the axis, you end up actually getting this thing that looks quite a bit. Um, like if you had just have, like, one over sign, which is CO c can. So the nice thing about this is that technically, at this point, it is that undefined. But there's this really cool thing is that you can actually take, um, you can take an approximation to get pretty close. And so if you zoom in, you know, pretty far you get something that looks about like 0.31 ish, which is pretty close. And so that's you can you can look at that and you might go Well, what is what is this 0.31 coming from? And so that's kind of like what part they want you to do, right? Look at this graph. Kind of see what's going on, and you end up getting something likewise your one and then and then investigated. Try to find out what's going on, which is what part B is so part be asked to you to kind of like, Hey, take your take your ah little walk down a number line rights and I want you. You know, you said that the 0.31 was in there right at zero. And these are X values. And then, you know, these are like our limits stuff, right? And then if you plug in stuff that's close, then this is sort of what what the game is. You put it in a 0.5, you plug in a 0.1, you get some numbers. You know, you might get like point for seven, and then you get like 70.38 and you get closer and closer and closer, and you you get something that feels a little bit like 0.31 Here's the interesting thing about this is that you might be wondering, Where is this 0.31 coming from? Well, it's much more than 0.31 It's actually if you take on, we get more into this and calculus, but actually, if you take this one from inside of the sign and the pie from inside of the sign and you get one divided by pi. One divided by pi is very close to 10.31 and that is not a coincidence. That's not exactly what this problem is asking you, but I think it's illuminating as to what you're going to be doing pretty soon. In the world of calculus, once we are done with limits, I hope this was helpful right until that time.

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