What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)?

Question

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)?

I took the numerator, used the change of coordinates $u = \alpha t$ and got that the limit was $\frac{1}{\alpha}$..This seems wrong but I'm not sure why. Any help would be appreciated!

Answer 1

$$\frac{\Gamma(\alpha t)}{\Gamma(t)} = \frac1{\alpha} \frac{\alpha t \Gamma(\alpha t)}{t \Gamma(t)} = \frac1{\alpha} \frac{\Gamma(1+\alpha t)}{ \Gamma(1+t)}$$

Now take the limit as $t \to 0$...