I apologize for being late on this thread. The limit in question does not require the full fire-power of Wolfram Alpha, adequate though that might be: I would expect a Calculus I student to be able to figure it out.
The only non-obvious fact is (sin h)/h->1 as h->0, which you need so that you can differentiate sin and cos using the definition. With this in mind, and remembering sin x=cos ((pi/2)-x), etc.: (x/4)tan (pi/2)(1-x)=(x/4)(sin (pi/2)(1-x))/(cos (pi/2)(1-x)) =(x/4)(cos (pi/2)x)/(sin (pi/2)x). As x->0, the cosine term goes to 1, and x/(sin(pi/2) x)->1/(pi/2) using the fact above. This gives the result. Best wishes, John Jose Mario Quintana wrote: > I asked WolframAlpha the following: limit of ((x/4) tan((pi/2)(1-x))) as > x-> 0 and it replied: 1/(2 Pi) (together with a nice graph). > > > On Wed, Feb 27, 2013 at 10:41 AM, Raul Miller <rauldmil...@gmail.com> > wrote: > >> On Wed, Feb 27, 2013 at 7:30 AM, Aai <agroeneveld...@gmail.com> wrote: >> >> (*:100)&* @: area _ >> >> 0 >> >> No, unfortunately J does not interpret the above sentence in that >> sense. >> > >> > If I'm not wrong then J is right about this one: >> > >> > pi(n-1) pi 0 >> > tg ------- tg ---- >> > 2 n 2 0 >> > lim -------------- = ---------- = ---- = 0 >> > n -> oo 4 n oo oo >> > >> > But the accuracy gives us a much earlier decline to zero, because >> ... >> >> It's usually a mistake to use infinity in calculations - infinity is >> an inconsistent number so you should expect inconsistent results when >> it is used in calculations. (Something related often happens when >> reasoning about division by an unknown sum.) >> >> Infinity is mostly convenient way of indicating neglect and, ideally, >> focussing the conversation elsewhere. >> >> -- >> Raul >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
