Is it OK to set something like : $$\lambda := \lim _{x\rightarrow x_0}$$
and then use $$\lambda( f_n)$$
Is it OK to set something like : $$\lambda := \lim _{x\rightarrow x_0}$$
and then use $$\lambda( f_n)$$
I think, a more conventional way would be to define a mapping $$\lambda: (\mathbb{R} \to \mathbb{R}) \to \mathbb{R}$$ with $$\lambda(f) = \lim_{x\to x_0} f(x).$$
Edit: as Robert Israel pointed out, one should of course make clear that the mapping is only defined for functions where the limit exists. So $$\lambda: D \to \mathbb{R}$$ where $D$ is the subset of $\mathbb{R} \to \mathbb{R}$.