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Suppose that $f(x)$ has a horizontal asymptote. Must it be the case that the derivative approaches $0$ as $x$ tends to infinity? I do not think so, and I think I have a counter example, but I have yet to prove it.

Of course, I know that the converse is not true (a derivative approaching $0$ need not come from a function with a horizontal asymptote... think $\ln x, \sqrt x$, etc).

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2 Answers

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Suppose $f(x)=\frac{\sin(x^2)}{x}$. Then, surely we have $\lim_{x\to \infty}\frac{\sin(x^2)}{x}=0$.

However, the derivative $f'(x)$ is given by $f'(x)=2\cos(x^2)-\frac{\sin(x^2)}{x^2}$ and $\lim_{x\to\infty }\left(2\cos(x^2)-\frac{\sin(x^2)}{x^2}\right)$ does not even exist.

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If $~f(x)$ is a function that is differentiable on the real line and $~y=a$ is a horizontal asymptote of $f(x)$, then can we say that $~y=0$ is a horizontal asymptote of $~f'(x)$?

I think this is true. Any help is appreciated.

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