$\lim\limits_{n \to \infty}{\lfloor x \rfloor + \lfloor x^2 \rfloor + ... + \lfloor x^n \rfloor\over x^n}, 1$$\lim\limits_{n \to \infty}{\lfloor x \rfloor + \lfloor x^2 \rfloor  + ... + \lfloor x^n \rfloor\over x^n}, 1<x\in \mathbb R^+$$
I have a problem solving this limit.Here is my solution. Let x be an integer, then we have:
$$\lim\limits_{n \to \infty}{ x + x^2 + ... + x^n\over x^n} = \lim\limits_{n \to \infty} {(x^n -1)x \over (x - 1) x^n } = \lim\limits_{n \to \infty} {\frac{x^{n+1}}{x^n}-\frac{x}{x^n}\over \frac{x^{n+1}}{x^n} -\frac{x^{n}}{x^n}} = \lim\limits_{n \to \infty} {x - \require{cancel} \cancelto{0}{\frac{x}{x^n}}\over x -1}= {x\over x-1}$$
But I don't know how to solve for non-integer x.

You proved that if $x \in \mathbb{N}_{\geq 2}$ the limit is $\frac{x}{x-1}$, but we can actually prove that for any $x>1$ we have $\lim\limits_{n \to \infty} \frac{x+\ldots+x^n}{x^n} = \frac{x}{x-1}$.
$x>1$, then $(x+\ldots+x^n) -n\leq\lfloor x \rfloor + \ldots + \lfloor x^n \rfloor \leq x + \ldots +x^n$, which show us that the limit is again $\frac{x}{x-1}$ (Why?)
