Can someone give me an example of a function that is decreasing monotonically,converges to $0$ as $x\to \infty$,but its derivative diverges as $x \to\infty$.I have an idea of how to make such a function but I cannot give its explicit form.I will construct a smooth functions with certain stair line sudden slants,and the slants each time will be steeper each time,It will roughly look like this
Although sketch is very rough,I think I could convey my idea,constructing this way,I can guarantee that the derivative oscillates infinitely(unboundedly above),but function gradually approaches the value $0$.But I could not find any explicit form so that I could express my idea in a concrete form.I had seen a very construtive type function in the site mathcounterexamples.net from where I found this question too.But I was looking for a simple example.Is there any,can anyone help me get such a function or Can I guarantee that such a stairlike function can be constructed without explicitly constructing one?
Best Answer
A PATH
If $\lim_{x\to +\infty} f'(x) = \pm \infty$, then $\lim_{x\to +\infty} \frac{f(x)}{x} = \pm \infty$, by de l'Hopital's rule.
However, you can easily construct examples where the function monotonically tends to zero with unbounded derivative, when $\lim_{x\to +\infty} f'(x)$ does not exist.
Take, e.g.
$$t(x) = \begin{cases} 1-|x| & (|x|\leq 1) \\ 0 & (|x| > 1)\end{cases}$$
and define
$$g(x) = \sum_{k=1}^{+\infty} (2k-1) t\left((2k-1)k^2(x-2k+1)\right).$$
If you sketch it, you'll see repeated triangular shapes with increasing altitude and decreasing base, so that the area is "under control", what is the area of each triangle?
Then analyze:
$$f(x) = \frac{\pi^2}{6}-\int_0^x g(t) dt$$.