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.

Answer 1

Let \[ \epsilon >0\] and let
\[ \delta =\frac {\epsilon}{2} \]
If \[ |x| \le \delta , \quad |y| \le \delta\]
then
\[| f(x,y)- f(0,0)| =|x|+|y| \le 2 \delta =\epsilon \]