Given any $\alpha > 0$ and $c > 0$, and $x \in \mathbb{R}$, does the following statement hold as $x \rightarrow \infty$? If true, is there a way to prove it ?
$$ x^{\alpha} \sim (x+c)^{\alpha} $$
where $\sim$ means that given functions $f(x)$ and $g(x)$, we have $f(x) \sim g(x)$ if $f(x)=O(g(x))$ and $g(x)=O(f(x))$.
This is equivalent to showing that the following limit is equal to $1$: $$\lim_{x\to\infty}\frac{(x+c)^\alpha}{x^\alpha}$$ But this can be rewritten as: $$\lim_{x\to\infty}\left(1+\frac{c}{x}\right)^\alpha,$$ which clearly tends to $1$ for any $c,\alpha>0$.