f(x,y)=|x|+|y| for all (x,y) belonging to R^2
Show that for the above function lim (x,y)-(0,0) for f(x,y) , exists using epsilon-delta method.

Question

f(x,y)=|x|+|y|     for all  (x,y) belonging to R^2
Show that for the above function lim (x,y)->(0,0) for f(x,y) , exists using epsilon-delta method.

anonymous · Answer

Assume that $$\lim_{(x,y) \rightarrow (0,0)} f(x,y) = |x| + |y| = 0$$, then 
$$||x|+|y| - 0 | < ε\:(1)$$ and $$| x - 0| < δ(ε)\:(2)\: , \: |y - 0| < δ(ε)\:(3)\:,\:\:for\:\:\: ε,δ > 0$$ Sum (2) and (3) and you see that for small values of ε you have respectively small values of δ(ε). So the limit is zero.

anonymous · Answer

I've attached the proof in a word document. I was having trouble with OCW's equation editor. I hope this helps.