I'm trying to show that as $\alpha$ tends to 0, the gamma distribution $\Gamma(\lambda,\alpha),$ is properly standardised, tends to the standard normal distribution. I have figured out that the moment generating function for the gamma distribution is $\left(\frac{\lambda}{\lambda-t}\right)^\alpha.$ Also, I've worked out that the mean and variance of a gamma random variable is $\frac{\alpha}{\lambda}$ and $\frac{\alpha}{\lambda^2}$ respectively.
However, I am not sure how to proceed further. I tried by defining $Z=\frac{X-\frac{\alpha}{\lambda}}{\frac{\alpha^0.5}{\lambda}}$ and using the fact that $M_{_Z}(t) = e^{bt}M_{X}(at)$
However, I can't show that $M_{_Z}(t)=e^{t^2/2}$ which is the moment generating function of a standard normal random variable. Is this the correct way to proceed?