Question about graph of sin function

Question

There is question I had on my mind for 3-4 years. If you have a function:- f(x) = (sin(x))^n If you increase the value of n slowly the Value of the function at all points, except where it is one, will approach towards zero. And at a value of n around a million, the function changes to almost an impulse at odd multiples of 90° and thus seems to be non differentiable at those points(I plotted the function in MATLAB). I want to know is the function differentiable at odd multiples of 90° degrees when n is very large ( tending to infinity). If yes, how the differentiability changes as n grows. Can I get an expression of the slope of the graph having n and x as variables? Thanks

Answer 1

The function is differentiable (infinitely often) for each $n$. Only the limit is not.