> In article > <email@example.com>, > WM <firstname.lastname@example.org> wrote: > >> On 9 Mai, 21:36, Virgil <vir...@ligriv.com> wrote: >> > In article >> > <b15d6323-e22a-4963-9519-e7f9e948e...@q8g2000vbl.googlegroups.com>, >> > >> > ?WM <mueck...@rz.fh-augsburg.de> wrote: >> > > > WM <mueck...@rz.fh-augsburg.de> wrote: >> > > > For all n: f(n) = 1 , lim_n-->oo f(n) = 1 >> > > > This is required for correctly calculating differential quotients >> > > > in analysis. (Just this morning I explained that in class.) >> > >> > How is >> > ? ?"For all n: f(n) = 1 , lim_n-->oo f(n) = 1" >> > needed ?to calculate ?the differential quotient of f(x) = e^x at x = >> > pi? >> >> It is necessary to calculate the differential quotient of functions >> like f(x) = ax + b. > > It is not at all necessary, as many calculus texts manage quite nicely > to find the differential quotients of such linear functions without it. >> > >> > It can ONLY be of any use in correctly calculating differential >> > quotients in the rare cases in which the difference quotients at a >> > point are all equal regardless of the differences in x. >> >> So it is. But even these "rare cases" belong to mathematics and have >> to be solved correctly. > > I do not know of any such rare cases that cannot be solved much more > simply by ordinary difference quotiens, >> > >> > I.e., when the delta-y over delta-x ratio is constant, as in linear >> > functions. >> > >> > So apparently WM never gets anywhere beyond the derivatives of linear >> > functions. >> >> That is not an admissible conclussion. > > Why not? Your sequential arguments are like using sledgehammers to crack > eggs.
Not really. They are the way you go if you unwind the definitions.
> Give y = f(x) = a*x + b for all real x, and son point x = x_0 > The difference quotient from x = x_0 to x = x_0 + h, for any h =\= 0 is > [f(x_0 + h) - f(x_0)]/[(x_0 + h) - (x_0)] = > [a*(x_0 + h) + b - a*x_0 - b]/h = > [a*h]/h = > a. > Since this is true for all non-zero real h, > it is also the limit as h -> 0. > No sequences needed. > > Is that too difficult for your students, or just too difficult for you?
Well, in your above computation, you still need to take lim_(x->x_0), and the final a is to be seen as a function with constant value a, for lim_(x->x_0) a to make sense. And the limit of that function at x_0 is taken through sequences, and that boils down to something remotely resembling Mückenheim's crap. One time at least you (resp. students) should see that and how the machinery works before starting to be sloppy. Or imagine programming a CAS system doing this stuff. You will need to distinguish the number a from the constant function with value a, and probably you will need to think a little bit to get things straight at this point. Mückenheim-style crap like For all n: f(n) = 1 , lim_n-->oo f(n) = 1 is necessary to calculate the differential quotient of functions like f(x) = ax + b. will make a mess out of your CAS. It does make some sense that the f(.) notation is used for functions, and the index notation (x_i) is used for sequences, but Mückenheim proudly screws everything up.
But this is just the first level of current Mückenheim crap. This strawman about differentiating linear functions was brought in to start a new round of Mückenheim's idiotic digit shuffling procedure allegedly proving that in what Mückenheim believes to be set theory, "For all n: f(n) = 1 , lim_n-->oo f(n) = 0".