On 8 Mai, 21:56, Virgil <vir...@ligriv.com> wrote: > In article > <15ddac8c-14be-485b-bac7-213b078c1...@k8g2000vbz.googlegroups.com>, > > 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.) > > My sympathy for your poor students to be subjected to such incompetence. > > Curious that that particular limit appears so rarely in calculating > differential quotients or in calculus texts. > > One does not find it referred to at all in such calculus texts as > Apostol. > --
If for every sequence (x_n) with limit x_0 the limit of the sequence of difference quotients (f(x_n) - f(x_0))/(x_n - x_0) exists and is the same in all cases, then df/dx is defined at x_0.
The function f(x) = x has for every sequence x_n --> x_0 the sequence of diference quotients (x_n - x_0)/(x_n - x_0) = 1. The limit is 1 in all cases, and not 0.