Weak and strong convergence of operator sequence

Question

Let $|e|\leq1$ such that $e\neq0$, we define the lineal operator $F_e: C^1([-1,1],R)\rightarrow R $ as $$ F_e x(t) =\frac{x(e) - x(-e)} {2 e} . $$ I need to know if the sequence of operators ${\{F_e\} }_e$ converges weakly and/or strongly to some operator as $e\rightarrow0$. I suspect that limit could be $x'(0)$ but Idk how to prove it or disprove it.