Prove that $\lim_{x \to \infty} f(x) = 0$

Asked Dec 23 '16 at 14:37

Active Dec 23 '16 at 14:37

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Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and differentiable function. If $\lim_{x \to \infty} ((f'(x))^2 + f^3(x)) = 0$, prove that $\lim_{x \to \infty} f(x) = 0$.

Here is what I did so far:

$\lim_{x \to \infty} ((f'(x))^2 + f^3(x)) = (\lim_{x \to \infty} f'(x))^2 + (\lim_{x \to \infty} f(x))^2 \cdot \lim_{x \to \infty} f(x) = 0$

From this, we can conclude that $\lim_{x \to \infty} f(x) \leq 0$.

Now, I need to prove that $\lim_{x \to \infty} f(x)$ can't be negative, but I didn't figure out how to do this yet.

Thank you!

asked Dec 23 '16 at 14:37

George R.

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Why do these limits exist? – Paolo Leonetti Dec 23 '16 at 14:39
@Mathematician42 I highly doubt that $f^3(x)=f(f(f(x)))$ – Mark Viola Dec 23 '16 at 14:45
@Mathematician42 your usual is very context specific. In analysis context basically no one reads $f^n$ as $n$ times composition of $f$ with itself. – user251257 Dec 23 '16 at 15:09

Prove that $\lim_{x \to \infty} f(x) = 0$

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