Re: [math-fun] mis-defining limits
<< I'm teaching an honors calculus class this semester, and to motivate the formal epsilon-delta definition of limits, I'd like to show the students some appealing, informal definitions of limits and the hidden errors and ambiguities of those definitions. Do any of you have any favorite examples of this?
If you can first use (informal) limits to show that the slope of the tangent line T_x to y = x^2 at (x,x^2) is 2x, then an interesting limit is the limit of the y-intercept of T_c as c -> 0. A similar example is the limit of the area of the triangle formed by the normal line N_c to y = x^3 at (c, c^3), and the x- and y-axes, as c -> 0+. << E.g.,, I seem to recall that some definitions of limits say that f(x) should approach but never equal b as x approaches a (which would make lim_{x -> 0} x sin x problematical).
To say nothing of limits of constant functions! I've never heard this condition, but such a notion is SO WRONG that I wouldn't even put it into student's heads. *Especially* since there is a very slightly related condition that IS part of the definition of a limit: Given f: R -> R, say, the (limit of f(x) as x -> a) = L means (and of course you know the following; it's just the way I like to define a limit when we finally get around to the formal def. -- in four easily understood parts: ---------------------------------------------------------------------------------------- For all e > 0 there exists a number d > 0 (that may depend on e) such that if 0 < |x - a| < e then |f(x) - L| < e. ------------------------------------------------------------------------------------------ It is the lower left condition, in particular the 0 < |x - a| part of the inequality, that a student could easily confuse with the very wrong condition "but never equal b" that you mention, Jim. I hope you never once bring it up. Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . . (which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators. --Dan
On 9/10/06, Daniel Asimov <dasimov@earthlink.net> wrote:
Btw, on his website, Noam Elikies asks what the limit is of the infinite series
g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . .
(which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
--Dan
There was a series of slim books by Konrad Knopp entitled "Counterexamples in ..." --- the "Analysis" volume would no doubt give more relevant examples. The one quoted is an example of a function of a complex variable with a continuous frontier of singularities around the unit circle, so incapable of analytic continuation by standard infinite sum techniques --- I don't know whether continued fractions might be more successful? Recently, in the course of attempting to blend two functions along an interval while preserving all derivatives at the endpoints, I encountered a function with continuous derivatives of all orders on the real line, yet nowhere analytic there. [This horror may well constitute the frontier of some function analytic on the upper half plane, which I was unable to determine.] Anybody know of anything similar, but perhaps simpler? WFL
and they said ``why is that so hard?''
Has anyone ever taught the nonstandard analysis version? -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
Is there any difficulty as x --> 0- or 0+ ? I thought that the interesting question is x --> 1- R. On Mon, 11 Sep 2006, Fred lunnon wrote:
On 9/10/06, Daniel Asimov <dasimov@earthlink.net> wrote:
Btw, on his website, Noam Elikies asks what the limit is of the infinite series
g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . .
(which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
--Dan
There was a series of slim books by Konrad Knopp entitled "Counterexamples in ..." --- the "Analysis" volume would no doubt give more relevant examples.
The one quoted is an example of a function of a complex variable with a continuous frontier of singularities around the unit circle, so incapable of analytic continuation by standard infinite sum techniques --- I don't know whether continued fractions might be more successful?
Recently, in the course of attempting to blend two functions along an interval while preserving all derivatives at the endpoints, I encountered a function with continuous derivatives of all orders on the real line, yet nowhere analytic there. [This horror may well constitute the frontier of some function analytic on the upper half plane, which I was unable to determine.]
Anybody know of anything similar, but perhaps simpler? WFL
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