Delta-Epsilon Proofs and Arbitrary Epsilon ChoiceDate: 11/05/2002 at 11:52:32 From: Morris Waehlti Subject: Do delta-epsilon proofs have to seem so arbitrary ? Dear Doctor Math, The limit of f(x)=x^2 + 1 as x approaches 2 is 5. Now a rigorous proof of this would seem to involve showing that for any choice of epsilon, the corresponding value for delta is min{1, epsilon/5}. Just what is this rigorous proof? How is delta = epsilon/5 obtained? More generally, what is the nature of the special relationship between a function's behavior near a point, the choice of epsilon, and a corresponding "good" delta? I hope that this is not too much to ask. Very best regards, Morris Date: 11/05/2002 at 13:04:26 From: Doctor Peterson Subject: Re: Do delta-epsilon proofs have to seem so arbitrary ? Hi, Morris. There are actually many ways you can choose delta; the choice of 1 in your answer is arbitrary, and any delta less than what you show will be satisfactory. There is no need to find the "best" delta, only one that will guarantee that the resulting error will be less than epsilon. In fact, there are entirely different ways to choose delta for this example, as you'll see in the second link below. I think a general answer to your general question is that the bounds on the slope of the function in the neighborhood of the point under consideration will play a large role in determining what delta will look like. We have several discussions of delta-epsilon proofs on our site, both explaining the general idea, and how to come up with deltas: Epsilon/Delta Definition of Limits http://mathforum.org/library/drmath/view/53738.html A Limit Proof Using Estimation http://mathforum.org/library/drmath/view/53357.html Formal Definition of a Limit http://mathforum.org/library/drmath/view/53403.html Understanding the Need for Limits http://mathforum.org/library/drmath/view/53754.html I think these should answer your questions well. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/06/2002 at 10:36:03 From: Morris Waehlti Subject: Do delta-epsilon proofs have to seem so arbitrary ? Dear Doctor Peterson, Thank you for your comments and also for the library references, which I have already printed out but not yet read completely. Concerning this business about, "the bounds on the slope of the function in the neighborhood of the point under consideration will play a large role in determining what delta will look like," this is what I am thinking at this point: Let a particular situation be: the limit of f(x) = x^2+1 as x approaches 2 is 5 with epsilon = 1 and delta = 1/5. Now the neighborhood of the point 2 is (9/5,11/5) and the corresponding neighborhood of 5 is (4.24,5.84). The slopes of f corresponding to the endpoints of the neighborhood of 2 are 3.6 and 4.4. So 3.6*1/5 = 0.72 and 5-4.24 = 0.76 are nearly the same. Also 4.4*1/5 = 0.88, which is not far from 0.84. Has this got something to do with what it's about? Propagation of errors through functions? Very best regards, Morris Date: 11/06/2002 at 12:03:35 From: Doctor Peterson Subject: Re: Do delta-epsilon proofs have to seem so arbitrary ? Hi, Morris. Yes, what you describe is what I had in mind. More precisely, the ratio of epsilon to delta will be, at least in "nice" cases like this, no more than the slope of the steepest chord from the limit point to points in the neighborhood. Because the parabola under consideration is smooth, with its slope either increasing or decreasing as you move away, this steepest chord will be at an endpoint of the interval: R +-------------------+ f(2)+e = f(2+d) | /| | / | | / | | / * | | / |e | / | | / * | | / | | Q/* | +---------+---------+ f(2) | /* | | | / * | | P+ * - - - | - - - - | f(2-d) | | | | | |e | | | | | | | | | | | | +---------+---------+ f(2)-e | d | d | 2-d 2 2+d In this example, we can just find d (delta) for a given e (epsilon) by solving f(2+d) = f(2)+e and then showing that, because of the way f curves, f(x) will be within e of f(2) whenever x is within d of 2. The ratio e/d is the slope of chord QR, which is between the slope of f at Q and the slope at R. Other functions are not as simple, and we can't find the greatest possible delta so precisely; but we approximate this by making various simplifications. You could probably, in many cases, at least make a good guess at delta based on slope considerations, and then proceed to prove your conjecture algebraically. Of course, you realize that there is not just one delta we can use; any delta LESS than the "ideal" delta we found will satisfy the definition of a limit. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/06/2002 at 13:34:58 From: Morris Waehlti Subject: Thank you (Do delta-epsilon proofs have to seem so arbitrary ? ) Thank you very much, Doctor Peterson, for your help. Morris |
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