Problem 642 (difficulty: 2/10)

Let \(\displaystyle f:\R\to \R\) be differentiable, \(\displaystyle \lim_{x\to \infty} f=1\). Does it imply that \(\displaystyle \lim_{x\to \infty} f'=0\)? And if we also know that \(\displaystyle \lim_{x\to \infty}f'\) exists?

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