If $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim f'=0$

Show that if $f\in C^1(\RR)$ and both $\lim_{x\to \infty} f(x)$ and $\lim_{x\to \infty} f'(x)$ exist, then $\lim_{x\to\infty} f'(x)$ must be zero.

Solution

1. Apply the mean value theorem on $[x, x+1]$.

2. The left-hand side tends to $0$.

$\lim_{x\to\infty}f(x)$ exists, so $f(x+1) - f(x) \to L - L = 0$.

3. Conclude $\lim_{x\to\infty}f'(x) = 0$.

$\xi_x \to \infty$ as $x \to \infty$, and $\lim_{x\to\infty}f'(x)$ exists (call it $b$); along the path $x \mapsto \xi_x$, $f'(\xi_x) \to b$.

4. Q.E.D.

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