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Question:

Let $f:[0,\infty)\rightarrow \mathbb{R}$ be differentiable and such that $\lim_{x\to\infty} f'(x) = L$. Then for every $c>0$, $\lim_{x\to\infty} f(x+c)-f(x)=cL$ and $\lim_{x\to\infty} f(x)/x = L.$

What i think that may work: using mean value theorem for the first proof and l'hospital rule for the second proof. But i can't write this rigorously. How do i write those things in a rigorous way?

1 Answers 1

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For every $x>c$ there is $c_x\in [x,x+c]$ such that $f(x+c)-f(x)=cf'(c_x)$, as $x\leq c_x\leq c+x$ then $c_x\to +\infty$ as $x\to +\infty$, hence $f'(c_x)\to L$ , so $f(x+c)-f(x)=cf'(c_x)\to cL$.

For the second: Hospital rule:

$\displaystyle\lim_{x\to \infty}\dfrac{f(x)}{x}=\lim_{x\to \infty}\dfrac{f'(x)}{(x)'}=\lim_{x\to\infty}f'(x)=L$

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    thank you, i was looking for a proper way to choose the interval in order to apply mean value theorem (without writing too much or getting confused). For l'hospital i must show that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$, but i can figure that using the last result. – 2017-02-08