With the first one I would probably go with the limit route:

Take , now, since both the numerator and denominator tend to infinity, we aply L'Hopital's rule and reach the equivalent which tends to infinity.

For the second one I don't see any problem with your argument if you already know (ie. have proven) that exponential growth is indeed faster than polynomial growth. At any rate the proof of that would be almost the same as the first one, only with instead of and the limit would then tend to zero (if you put the exponential in the denminator