 Solution

For every $n\ge 1$, $|x^n\sin (1/x)|\le |x{|}^n\to 0$, so the squeeze theorem gives limit zero. For $n=0$, sequences $x_k=(\pi /2+2k\pi {)}^{-1}$ and ${\tilde{x}}_k=(3\pi /2+2k\pi {)}^{-1}$ give function values $1$ and $-1$, so no limit exists. HenceThe value assigned at zero when $n=4$ does not affect this punctured limit.