Why are additive limit preserving functors left exact?

Question

I understand that the kernel is a limit and thus preserved by the functor, but why is exactness preserved in the middle? i.e How does the exactness of $0\to A\to B\to C\to 0$ imply the exactness of $0\to F(A)\to F(B)\to F(C)$ at $F(B)$, where $F$ is limit preserving?