Can you please check my proof of this limit of derivative

Hmm, I do have my doubts how you arrive at the conclusion $f^\prime(x) < \frac{L}{2}$. I get $f^\prime(x)< \frac{3L}{2}$ or $f^\prime(x) >\frac{1}{2}$ Also, assuming you can show that, you should at least explain why you may assume that $L\neq 0$, since you make explicit use of that fact (this is of course easy, you can, e.g. just add $ax$ to $f$)

To be honest, I also think the approach is too complicated. I hope you don't mind if I propose an easier approach: by the mean value theorem, for each $x$ near $0$ there is $c\in (0,x)$ (or $\in (x, 0)$, depending on the sign of $x$), such that $$\frac{f(x)-f(0)}{x} = f^\prime(c)$$ Now if you let $x\rightarrow 0$ you get the claim immediately from the definition of differentiability and the assumption about the convergence of $f^\prime$ (since, of course, the corresponding $c\rightarrow 0$, as well).

This path of thought actually incited me to write my first comment, now deleted (which said you need to show first that $f^\prime(0)$ exists). With this reasoning you don't need to know that $f$ is differentiable at $0$, you get it for free.