1. ## Powers of limits

Given that
$\displaystyle e=lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n$

Wikipedia's proof that (positive integer $\displaystyle x$)
$\displaystyle e^x=lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n$

contains this step:
$\displaystyle \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left( \left(1+\frac{1}{n} \right)^n \right)^x$

Is that allowed? Why?
Thanks

Edit
$\displaystyle \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdot lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdots lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n (?=) lim_{n \to \infty} \left(1+\frac{1}{n} \right)^{nx}$

2. Originally Posted by MSUMathStdnt
Given that
$\displaystyle e=lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n$

Wikipedia's proof that (positive integer $\displaystyle x$)
$\displaystyle e^x=lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n$

contains this step:
$\displaystyle \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left( \left(1+\frac{1}{n} \right)^n \right)^x$

Is that allowed? Why?
Thanks

Edit
$\displaystyle \left(lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \right)^x = lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdot lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n \cdots lim_{n \to \infty} \left(1+\frac{1}{n} \right)^n (?=) lim_{n \to \infty} \left(1+\frac{1}{n} \right)^{nx}$

Because the exponential function is continuous, thus $\displaystyle \displaystyle{\lim\limits_{n\to\infty}(a^n)^x=(\li m\limits_{n\to\infty}a^n)^x\,,\,\,0<a\in\mathbb{R}$

tonio