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If both the limits $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim_{x\to\infty}f'(x)=0$.

**Question.** Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function. If both the limits $\displaystyle \lim_{x\to\infty}f(x)$ and $\displaystyle \lim_{x\to\infty}f'(x)$ exist, how does one show that $\displaystyle \lim_{x\to\infty}f'(x)=0$?

**Note.** This statement is intuitively obvious: if $f$ has a horizontal asymptote, then the slope of the graph goes to zero as the graph approaches the asymptote. 

It seems to me that proving by contradiction is easier to work with than a direct proof where one needs to estimate the size of the derivative. I will write my own answer below. Other perspectives are welcome.