How am I looking at this Limit wrong?

Here's the limit I'm working with:

$\displaystyle \displaystyle\lim_{x\to\infty}(1 + \frac{2}{x})^x$

Now, I understand that the correct answer is $\displaystyle e^2$. While I don't fully understand how you get there, I understand that you can manipulate this until you come up with $\displaystyle e^2$.

However, here's how I'm looking at it...

As x approaches infinity, the expression within the parentheses approaches 1. Which to me, means that the limit of this whole thing is $\displaystyle \displaystyle\lim_{x\to\infty}(1)^x$, which is one.

So, my question is - how is the way I'm conceptually looking at this incorrect?