# $\lim\limits_{x\to\infty}f(x)$ exists but $\lim\limits_{x\to\infty}f'(x)$ does not

The following problem is from Calculus by Spivak:

Give an example of a function $f$ for which $\lim\limits_{x\to\infty}f(x)$ exists, but $\lim\limits_{x\to\infty}f'(x)$ does not exist.

Presumably, $f$ should also be differentiable. I manage to visualize such a function, but the definition is rather messy:

Let $s_0=0$ and $s_n=\sum\limits_{i=1}^n\frac1n$ for any positive integer $n$. Define $$f(x)=\begin{cases}x&\text{if }x<0\\(-1)^{n+1}\frac{\sin n(x-s_{n-1})}n&\text{if }x\in[s_{n-1},s_n)\end{cases}$$ I'm looking for simpler functions that satisfy the conditions. Feel free to give more than one example.

• What about $\sin(x^2)/x$? – Aleksey Pichugin Nov 6 '11 at 10:58

## 1 Answer

Take $f(x)=\frac{\cos \left(x^2\right)}{x}$ for example: It clearly goes to $0$ but the derivate keeps somewhat nasty for high $x$.

Here is a plot of $f'(x)$: • Here's wolframalpha link. – Martin Sleziak Nov 6 '11 at 10:58
• Thanks! I was playing around with functions like $sin(x)/x$, $sin(1/x)/x$, etc, but couldn't figure out the appropriate one. – Ben Nov 6 '11 at 11:04