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?