Azhaius
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1. Find the horizontal asymptotes of the graph of the function $$g(x) = \frac{|f(x)|}{x-2}$$ if $$f(x)$$ satisfies inequality $$f(x+1) \le x \le f(x)+1$$ for every real $$x$$.
2. Let $$f: R->R$$ be a function that is differentiable at zero and such that $$f(0) = 0$$. Show that for each $$n\in\mathbb{N}$$ we have that
$$\lim_{{x}\to{0}} \frac{1}{x} ( f(x) + f(\frac{x}{2}) + ... + f(\frac{x}{n}) ) = (1 + \frac{1}{2} + ... + \frac{1}{n} ) f'(x)$$
2. Let $$f: R->R$$ be a function that is differentiable at zero and such that $$f(0) = 0$$. Show that for each $$n\in\mathbb{N}$$ we have that
$$\lim_{{x}\to{0}} \frac{1}{x} ( f(x) + f(\frac{x}{2}) + ... + f(\frac{x}{n}) ) = (1 + \frac{1}{2} + ... + \frac{1}{n} ) f'(x)$$
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