# Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: $$\lim_{(x,y)\rightarrow(0,0)}\frac{y^2+3xy+x^2-2x-3y+1}{\sqrt{(x-1)^2+y^2}}$$ and it would be great if you could give me some tips for future limits as such, thanks! (:

• You want that $1$ in the denominator? – zhw. Jun 27 '15 at 19:16
• In this case, you can use the fact that the top and bottom both have limits of 1 as $(x,y)\to(0,0)$. – user84413 Jun 28 '15 at 0:24