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