Maybe you should be demonstrating that the MGF fails to exist? Maybe you could prove it for one variable? Maybe not.
Below I've attached the problem. The structure is:
1. Question asked
2. My Attempt
3. Last step (I've labeled this out to help clearly identify where my problem is).
I've double checked my work and I keep arriving to this final expression, which I believe is correct. Problem is, doesn't my expression explode since e^infinity explodes?