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Suppose we have function $f:\Bbb R^+\to\Bbb R$ that is bounded $|f(t)|<M$ and differentiable such that $\lim\limits_{t\to\infty}f'(t) = 0$. Does this imply that $f(t)$ converges?

Matcha Latte
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stove
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2 Answers2

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First idea: If you drop the boundedness assumption, $\ln(x)$ does provide a counterexample to what you want.

Now think about how you can transform this counterexample into a bounded counterexample.

Hint

Typical bounded functions include $\sin$ and $\cos$.

PhoemueX
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  • Thankyou so much! Ive spent way too long thinking about that... – stove Feb 03 '15 at 03:57
  • Sorry for being a bit pedantic, but you'll need $\ln(x+c)$ for some $c>0$ to get a counterexample. $\ln$ itself is not bounded on all of $\mathbb{R}_+$. (The bounded counterexample I came up with using your hint is neat, though!) – Ian Feb 03 '15 at 04:23
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In general the answer is NOT. Let's consider the function $f(t) = \sin(\ln(t))$. Its derivative $f' = \cos(\ln(t))/t$ converges to $0$ but in limit, the function keeps oscillating.

YuiTo Cheng
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Hung Tuan
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