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Let $f$ be a continous differentiable function on $\mathbb R$. Suppose that $$L = \lim_{x \to \infty} ( f(x) + f'(x) )$$ exists. If $ 0 < L < \infty$, then which of the following statments is / are true ?

  1. if $\lim_{ x \to \infty} f'(x)$ exists, then it is $0$.

  2. if $\lim_{ x \to \infty} f(x)$ exists, then it is L.

  3. if $\lim_{ x \to \infty} f'(x)$ exists, then $ \lim _{x \to \infty} f(x) = 0$ .

  4. if $\lim_{ x \to \infty} f(x)$ exists, then $ \lim _{x \to \infty} f'(x) = L$

Any help would be appreciated. Thank you

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merged by davidlowryduda Nov 26 '15 at 2:40

This question was merged with $L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct? because it is an exact duplicate of that question.