Let $f$ be a continuously differentiable function on $\mathbb{R}$. Suppose that $L=\lim_{x\to \infty}(f(x)+f'(x))$ exists.
If $ 0\lt L \lt\infty$ ,then which of the following statements is/are correct?
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 $\lim_{x\to \infty}f'(x) =L$
Let $f(x)=1+ 1/x$ then $\lim_{x\to \infty}f(x)=1$ and $\lim_{x\to \infty}f'(x)=0$ so $L=1$
using this example 1,2 correct but this function is not differentiable at x=0 so how to proceed..