I am trying to simplify this limit $\lim_{n \to +\infty}\left(1 + \frac{1}{n}\right)^{nx}=e^x$ into the well-known definition of $e^x$ as:
$$\lim_{n \to +\infty}\left(1 + \frac{1}{n}\right)^{nx}=e^x \iff \lim_{n \to +\infty}\left(1 + \frac{x}{n}\right)^{n}=e^x$$
I would prefer a clear explanation on how to transform the first expression into the second one without involving any other exponential limit definitions or logarithmic expressions, just using limits rules and algebra, and binomal expansion if necessary.
Thank you.
Best Answer
If $x>0$, substitute $y=nx\implies\frac1n=\frac xy$. Then as $n\to\infty$, we have $y\to\infty$:
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{nx}=\lim_{y\to\infty}\left(1+\frac xy\right)^y$$
If $x<0$, substitute $y=-nx\implies\frac1n=-\frac xy$. Then $y\to\infty$:
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{nx}=\lim_{y\to-\infty}\left(1-\frac xy\right)^{-y}$$
Now swap out the symbol $y$ for $n$ in the first case and $-n$ in the second.