Show that $\lim _{x \rightarrow \infty}\left|f(x)+f^{\prime}(x)\right|=L \in (0,\infty)$ implies $\lim _{x \rightarrow \infty} f(x)=L$

Question

Suppose that $f$ is a continuous differentiable function on $\mathbb{R}$ with $$\lim _{x \rightarrow \infty}\left|f(x)+f^{\prime}(x)\right|=L \in (0,\infty),$$ then, which of the following is/are true,

(1) if $\lim _{x \rightarrow \infty} f(x)$ exists, then it is zero.

(2) if $\lim _{x \rightarrow \infty} f(x)$ exists, then it is $\mathrm{L}$.

(3) if $\lim _{x \rightarrow \infty} f^{\prime}(x)$ exists, then $\lim _{x \rightarrow \infty} f(x)=0$

(4) if $\lim _{x \rightarrow \infty} f(x)$ exists, then $\lim _{x \rightarrow \infty} f^{\prime}(x)=L$

I will reject options 1 and 4 by taking the constant function $f(x)=1$, How can we deal with options 2 and 3? Can I assure that the limits $$\lim _{x \rightarrow \infty} f(x)~\text{and}~\lim _{x \rightarrow \infty} f'(x)$$exist always? I couldn't find any counters.

Thanks in advance.

Answer 1

2. is false: Take $f(x)=-L$.

3. is false too: Take $f(x)=1+e^{-x}$.