Question

can someone help me out with this limit?

lim(x,y)->(0,0) ((2xy(x+y)))/((x^(2)+y^(2)))

it goes to zero, but how?

Answer 1

\[\lim_{x,y \rightarrow 0,0} [(2xy(x+y))/(x^2+y^2)]\] Start by expanding to make the limits easier.
\[\lim_{x,y \rightarrow 0,0} [(2x^2y+2xy^2)/(x^2+y^2)\]
\[\lim_{x,y \rightarrow 0,0} [(2xy^2/(x^2+y^2))-(2y^3/(x^2+4^2)+2y]\]
Do you see how to do this limit now?