Final answer:
The existence of the limit lim->(0,0) of ((x²)y eⁿ²/(x⁴+4y²)) cannot be determined without further analysis, as initial examinations along simple paths (x-axis and y-axis) suggest the limit might be zero but more complex paths require a deeper examination.
Step-by-step explanation:
The question at hand is whether the limit lim->(0,0) of ((x²)y eⁿ²/(x⁴+4y²)) exists. To determine if the limit exists, one common method is to approach (0,0) along different paths and see if the limit values are consistent.
Let us first approach along the x-axis, so y=0. In this case, the limit simplifies to:
lim(x->0) of (0·eⁿ²/(x⁴)) = 0
Next, let's approach along the y-axis, so x=0. The limit again simplifies to:
lim(y->0) of ((0²)y eⁿ²/(4y²)) = 0
If we approach along y=x, we get a different scenario:
lim(x->0) of ((x²)x eⁿ²/(x⁴+4x²)) = lim(x->0) of (x³ eⁿ²/(5x⁴))
This limit does not simplify easily and suggests the existence of the limit may be more complex. Therefore, without further analysis, we cannot conclude unequivocally that the limit exists or not, as the function may behave differently along complex paths. More rigorous analysis or the application of certain theorems may be needed to determine the existence of the limit conclusively.