Given the piecewise function
g(x) =
{ x^(3)e^(-x^(2)/4)sin(4/x^(2)) if x=/=0
{ 0 if x = 0
Prove that g is differentiable and g' is bounded on (-infinity,infinity).
I know that I have to use the definition of a derivative for the first part (not too sure about how to do that). How to show it's bounded, I am not sure.
This is the first time I have ever seen anything like this.
Thanks for any help.
Pay attention to the fact that the derivative is an even function and thus it's enough to show boundness in, say ,:
But , , so the only problem is with the middle summand, but in fact:
(Applying L'Hospital to the first factor we get zero, whereas the second one converges to 1) , so the whole thing's bounded.
Tonio
Sorry, I actually mislead you the way I wrote that. I wrote thinking that it would fall under the category, but it doesn't because .
So the way you should actually do the limit is to use the squeeze theorem.
Thus, , so the limit is zero.
Sorry for any confusion I may have caused.