where the contour is $$|z|=1$$
if I use residual theorem $$res f(z) = \lim\limits_{z\to 0} z^3\cdot e^\left(\frac{1}{z}\right)\cdot \sin\left(\frac{1}{z}\right)$$
this limit is undefined .However , if i expand the function in its laurent series i can find coefficient of first negative power which is equal to $\left(\frac{2}{3}\right)$ How is this possible $?$
The "quick" formulae for residue only works for simple poles and poles "of finite order". They won't work for essential singularities though. However, Laurent series expansion always works.