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I came across this question in order to solve another problem: "If $|f'(x)|\leq M$, what can be said about $\lim_{x\to\infty} f(x)$?

I made an attempt to answer it, here it is:

By the mean value theorem, for every $(a,\infty)$ exists $c_x\in(a,x)$ such that:

$|f(x)|=|f(a)+f(c_x)(x-a)|\leq |f(a)+M(x-a)|$. Since $(x-a)\rightarrow \infty$ as $x \rightarrow \infty$, it follows that $\lim_{x\to\infty} f(x) = \infty$.

Is it correct?

  • 1
    That shows that the limit is bounded above by $\infty$, which doesn't help much. – 2017-02-09
  • 3
    No. Take $f(x)=\sin(x)$ for example. Or a constant $f(x)$ for that matter. – 2017-02-09
  • 0
    does it guarantee that $\lim_{x\to\infty} f(x)$ is finite, or at least bounded? – 2017-02-09
  • 1
    It doesn't guarantee anything. Consider $f(x) = x$. $f'$ is bounded and even constant but it diverges to infinity. – 2017-02-09

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It tells you the function is $\mathcal O(x)$ which imo is useful.

To see this notice that if $|f'(t)|\leq M$ for all $t$ then for positive $x$:

$|f(x)|=|\int\limits_0^x f'(t)dt+f(0)|\leq \int\limits_0^x |f'(t)|dt+|f(0)|\leq \int\limits_{0}^x Mdt+|f(0)|\leq Mx+|f(0)|$

  • 1
    $f'$ need not be Riemann integrable – 2017-02-09