MGF of n iid poisson rvs as n tends to infinity

Question

I am given the MGF of IID RVs as $ e^{\lambda(e^t-1)}$ and am supposed to find the MGF of their sample mean of n such RVs. I am arriving at

$M_n(t) = e^{n\lambda(e^{t/n} - 1)}$

This does appear to be a valid MGF, however, there is another part to the question where it asks for the $\lim_{n \to \infty} M_n(t)$

I am confused if it is proper to have the MGF equal to 1 for ${n \to \infty}$.

Thank You