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### Limit of $y$ if limit of $y+y'$ goes to $0$? [duplicate]

Let $y$ be a differentiable function on $\mathbb{R}$. If $$\lim_{x\rightarrow \infty}(y(x)+y'(x))=0$$ Then how does one show that $$\lim_{x\rightarrow \infty}y(x)=0$$ I'd appreciate some help on this ...
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### What can we say about $\alpha$ if $\lim_{x\to \infty}f(x)=1$ and $\lim_{x\to \infty}f'(x)=\alpha$ for a differentiable function $f$ on $\mathbb{R}$? [duplicate]

For what values of $\alpha$ would the given conditions hold? I don't know how to proceed.
### finite $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} f'(x)$. Then $\lim_{x \rightarrow +\infty} f'(x) = 0$ [duplicate]
Let f be twice differentiable function on R with finite $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} f'(x)$. Then $\lim_{x \rightarrow +\infty} f'(x) = 0$ I don't know how ...