Let $X \sim \operatorname{Poisson}(\lambda)$ and denote $\bar{X}=\frac{1}{\lambda}X$. By using mgf I want to show that $\bar{X}$ goes to $N(1,\frac{1}{\lambda})$ as $\lambda$ increases.
More precisely, I want to show that, for $Y\sim N(1,\frac{1}{\lambda})$, $$|M_{\bar{X}}(t)-M_{Y}(t)|\leq \frac{C}{\lambda^2}$$
So we need to show that $$|e^{t+\frac{t^2}{2\lambda}}-e^{\lambda(e^{t/\lambda}-1)}| \leq \frac{C}{\lambda^2}.$$
I tried Taylor expansion but I couldn't simplify it.