If $A$ is a Noetherian ring, $M$ is a finitely generated module, $I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know that $\hat{A}\otimes_{A}M\cong\hat{M}$.

Also in Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$ finitely generated. but the functor $M \mapsto \hat{A}\otimes_{A}M$ is always exact.

How to prove this assertion?

And what is an example of the breakdown of exactness of $M \mapsto \hat{M}$ when $M$ is not finitely generated?

(This is not a homework problem)