Prove $e^x$ limit definition from limit definition of $e$. Is there an elementary way of proving
$$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n,$$
given
$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n,$$
without using L"Hopital's rule, Binomial Theorem, derivatives, or power series?
In other words, given the above restrictions, we want to show
$$\left(\lim_{n\to\infty}\left(1+\frac1n\right)^n\right)^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n.$$

If you accept that exponentiation is continuous, then certainly
$$\left(\lim_{n\to\infty}\left(1+\frac1n\right)^n\right)^x = \lim_{n\to\infty}\left(1+\frac1n\right)^{nx}$$
But if $u=nx$, then by substitution we have
$$
\lim_{n\to\infty}\left(1+\frac1n\right)^{nx}=\lim_{u\to\infty}\left(1+\frac{x}{u}\right)^u
$$
