Find the result if the following limit exists. If not, find out why.

Question

lim(x;y)--> (0;0) x^3.y/x^6 + y^2

Answer 1

This limit does't exist, because if for example (x, y) → (0,0) along line y = x then lim x →0 of x³.x/(x⁶ + x²) = lim x²/(x⁴ + 1) = 0. If (x,y) → (0, 0) along line y = x³ then lim x → 0 of x³·x³/(x⁶ + x⁶) = 1/2.

So, lim as (x,y) → (0, 0) of x^3·y/(x^6 + y^2) DNE