Give an example of an infinitely differentiable function $h(x)$ with the same Taylor series as $\sin(x) $

Question

Give an example or counter example of an infinitely differentiable function $h(x)$ with the same Taylor series as $\sin(x) $but such that $h(x) ≠ \sin(x)$ for all $x ≠ 0$.

Answer 1

HINT: I think $$h(x) = \sin(x) + e^{-1/|x|} $$ with $h(0) =0$ is an example but I'll leave it to you to prove.