What is the lim x->0 (1+3x) ^ (1/x)

You can solve this using the definition of e: this might help you:

Exponentiation - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Exponentiation#Powers_of_e)

Think of n as used on this site as 1/x in your problem (i.e, this site talks about the limit as n approaches infinity, whereas you want the limit as x approaches 0). You'll see that your problem reduces to a power of e. Can you take it from here?

Can you use L'Hopital's method? Is that a legal move?

No, L'Hopital is probably not the best to tackle this one.

Use the fact that \lim_{x\to\0}(1+x)^{\frac{1}{x}}=e

Let t=3x and x=t/3.

\lim_{t\to\0}(1+t)^{\frac{3}{t}}

\lim_{t\to\0}[\underbrace{(1+t)^{\frac{1}{t}}}_{\text{this is e}}]^{3}

Now, see what you get as the limit?

Oh wow that's so much simpler. I used L'Hopital... still got the same answer but wow nty version was so much more complicated. Thank you very much!