I just thought I would throw this out here, hopefully it's not too boring of a question. I am stuck attempting to evaluate the following limit (these are all real numbers): $$ \lim_{\left(\chi_1, \chi_2\right) \to \left(\alpha_1, \alpha_2\right)} \frac{ \left(\left(\chi_1 + \chi_2 \right)^2 - \left(\alpha_1 + \alpha_2\right)^2\right) \hspace{1mm}-\hspace{1mm} \frac{1}{2} \left(\chi_1^2 + \chi_2^2 - \left(\alpha_1^2 + \alpha_2^2\right)\right) }{ (\chi_1 + \chi_2 - \alpha_1 - \alpha_2)^2 } $$
So, essentially, I seek to evaluate the derivative of the function
$ f : \mathbb{R}\times\mathbb{R} \to \mathbb{R} : (x_1, x_2) \mapsto (x_1+x_2)^2 - \frac{1}{2}\left(x_1^2 + x_2^2\right), $
however I don't have a lot of experience evaluating limits of functions with a domain besides $\mathbb{R}$. Does anyone have any ideas/suggestions? It's greatly appreciated.