Let for , let for .
Show that but that does not exist.
Lol if I could just "do it" I wouldn't be wasting my time writing this.
I took the first derivative, and have the following:
Now I took it two more times but it doesn't get me anywhere, I know there's something I'm not seeing, some help would be really nice...
Ok, so assuming that HallsOfIvy's derivative is correct (which there is no need to assume) then you want to see why doesn't exist. So, assume that it does. Then, it equals and so . I need to assume that the limit existed so that I could combine the limits in the last part and clear out that cosine. So, by assuming that our limit existed we've show that exists as well. But, look at where and
lol
If you want to see why it is "obvious"...
if , . So the denominator is ...
... and since sine is bounded, if then so...
...and this oscillates infinitly as just like oscillates as
:]
(oh dont hurt me, I can be rigorous I swear!)