If a function has a finite limit at infinity, does that imply its derivative goes to zero?

Solution 1:

The answer is: No. Consider $f(x)=x^{-1}\sin(x^3)$ on $x\gt0$. The derivative $f'(x)$ oscillates between roughly $+3x$ and $-3x$ hence $\liminf\limits_{x\to+\infty}\,f'(x)=-\infty$ and $\limsup\limits_{x\to+\infty}\,f'(x)=+\infty$.

Solution 2:

Take a function that is $0$ except in a small neighborhood of each positive integer; at $n\in\Bbb Z^+$ it has a smooth bump of height and width $1/n$ whose rising part has a maximum slope of $n$. This function is differentiable and has limit $0$ at infinity, but its derivative has no limit at infinity.

Solution 3:

There is a famous theorem known as Barbalat's lemma, which states the additional condition for $\lim_{x \to \infty} f'(x) = 0$. According to the lemma, $f'(x)$ should be uniformly continuous on $[a, \infty)$. In many applications, the uniform continuity of $f'(x)$ is shown by proving $f''(x)$ exists and is bounded on $[a, \infty)$.

(See Wikipedia https://en.wikipedia.org/wiki/Lyapunov_stability#Barbalat.27s_lemma_and_stability_of_time-varying_systems for the statement of Barbalat's lemma and its applications in stability analysis).