Prove the limit of a multivariable function using epsilon-delta definition
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If (x,y)∈R2 such that |y|≤|x|, then
|5x3−x2y2|≤5|x3|+x2y2≤5|x3|+2x2=x2(5|x|+2);
if |x|≤1, then x2(5|x|+2)≤7x2; taking any ε>0, we have 7x2<ε if |x|<ε/7–√. We have proved this: for every ε>0, if |y|≤|x|<min{1,ε/7–√}, then |5x3−x2y2|<ε.
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