Graphing LimitsDate: 09/12/2001 at 13:43:47 From: Justin Morrisroe Subject: Calculus-Limits I was hoping you could explain the concept of "limits" and how to read them from graphs, and why substituting the "c"# into the equation doesn't always work (i.e. lim (x^2-1)/(x+1) ) x-->1 the answer is 2, but I din't get 2, I got undefined. Help, please! Date: 09/12/2001 at 15:01:27 From: Doctor Peterson Subject: Re: Calculus-Limits Hi, Justin. Here's the basic idea of a limit: We have a function, which may or may not be defined at x = c. We want to know how it behaves near x = c, so we look at values of x closer and closer to c. Suppose the graph of y = f(x) looks like this: y | o : | o : | o : L+ - - - - - - - - - -?- - - - - - - | : o | : o | : +--------------------+-------------- x c The closer x gets to c, the closer y gets to L. Then we say that f(x) approaches L as x approaches c; that is, L is the limit of f(x) as x approaches c. For the sort of functions we usually work with, L is in fact the value of f(c), so this seems like a waste of effort. But in some very important cases (those for which this concept was invented), f(c) is not even defined, so the limit is the only way we can talk about this idea of closeness. In a sense, the limit "fills the gap" in the definition of the function, allowing us to assign a value to f(c) that makes the function "continuous." The trick is to define "closer and closer" precisely. We do this by seeing it as sort of a game, the sort of game you might have between two gamblers who don't trust one another. You say it's close - but how do I know that what you mean by "close" is really close enough? So I make a challenge: I'll believe that L is the limit, if for ANY definition of "close" I choose (that is, any distance I call "close enough"), you can give me a range around x = c within which the function is always that close to L. That's what the "epsilon-delta definition" of a limit means. If _you_ can keep the function as close to L as _I_ want it to be, I'll believe you. (It's sort of like having one kid cut the cake and the other choose the larger piece.) Now, I'm a little confused about your example; you probably typed something wrong. If I replace x with 1, I get 0/2, which is just 0. A more interesting problem is lim (x^2-1)/(x-1) x-->1 To find this limit, you have to simplify the expression so that it is defined for x = 1, and then substitute. Without simplifying, you get 0/0, which is indeterminate. (That is, it might have any value at all, and can't be determined without the additional information in the limit.) So this is an example where the function is not defined at x = 1, yet we can talk about its limit there. What we are doing is showing that the function is equal to a new function everywhere except at x = 1, so we can "fill the gap" by saying it is also equal to that function at x = 1. This value is the limit. I think you may find this discussion useful: Understanding the Need for Limits http://mathforum.org/dr.math/problems/josh.2.19.01.html Here's another point to consider: the term "limit" is probably the source of much of the confusion over this concept, because it doesn't really fit this setting. It was originally used, I believe, of the limit of a sequence, where the terms a[i] approach some "limiting value" when i becomes infinite. (For example, the sequence a[i] = 1/i approaches zero as i becomes infinite.) The essential concept is the same, but there it makes more sense to call it a limit, because the terms are going closer to that value but never go beyond it. It's better at first to talk about "approaching L" rather than "having limit L" for the sake of clarity. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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