Question

Check whether the limit of the function

f(x, y) = 4x²y/ x^10 +3y² exist as (x, y) ➡(0, 0)

Answer 1

$\displaystyle \quad\text{Consider the power curve }x=t, y=at^8 \text{ approaching the origin as }t\rightarrow 0^+.\\ \text{The limit along this curve can attain any value by varying the parameter a:}\\ \lim_{(x,y)\rightarrow(0,0)}f(x,y)=\lim_{(x,y)\rightarrow(0,0)}\frac{4x^2y}{x^{10}+3y^2}=\lim_{t\rightarrow 0^+}\frac{4t^2at^8}{t^{10}+3(at^8)^2}=\lim_{t\rightarrow 0^+}\frac{4at^{10}}{t^{10}+3a^2t^{16}}\\ \qquad\qquad\qquad\quad=\lim_{t\rightarrow 0^+}\left(\frac{t^{10}}{t^{10}}\times\frac{4a}{1+3a^2t^6}\right)=\lim_{t\rightarrow 0^+}\frac{4a}{1+3a^2t^6}=4a\\ \text{Thus, the multivariable limit does not exist.}$