Can someone please explain to me why the following identity is true? $\lim_{x \to \infty}\left(1 + \frac{a}{x} \right)^x = e^a$
(I'll make a notation $L$ that is equal to the limit above.)
One 'proof' I saw went something like this: $L = \lim_{x \to \infty}\left(\left(1 + \frac{a}{x} \right)^\frac{x}{a}\right)^a = e^a$
That can't be right... right? Because there really is nothing stopping me from saying $L = \lim_{x \to \infty}\left(\left(1 + \frac{a}{x} \right)^\frac{x}{a + 1}\right)^{a + 1} = e^{a + 1}$ but that's obviously not true.
Edit: I posted my own answer to this question, where I explain what got me confused:
► http://math.stackexchange.com...35491#35491