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.
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? --