TITLE: $\displaystyle \lim_{x\to 0} \frac{e^{2x}-1}{3x} $ help please?
QUESTION [6 upvotes]: So I have to find the 
$$\displaystyle \lim_{x\to 0} \frac{e^{2x}-1}{3x} $$
I first solved this by L'hopital and got $\frac{2}{3}$ but now I read carefully and it says in my book that I shouldn't solve this by L'hopital..any hints?

REPLY [2 votes]: Let $e^{2x}-1=y$ that yields
$$\displaystyle \lim_{y\to 0} \frac{2y}{3 \ln(1+y)}=\frac{2}{3}$$
because $\lim_{y\to0} (1+y)^{1/y}=e$
Sister. (It's the most elementary proof I know)