Proving what? That does not exist? That should be obvious: cos(1/x) is always between -1 and 1 and the denominator goes to 0 so that term does not converge. 2 sin(1/x) is always between -2 and 2 so it cannot "offset" the fact that cos(1/x)/x diverges.
i got a different result.
i assumed that in point x=0 the function turns to 0 too.
so in the plus side i will get 0
and on the minus side
but i cant keep doing this derivatives till i get two different limits
there must be an easier way
??