Question

Check whether the limit of the function
f(x,y)=3x^3y/x^6+2y^2 exists as (x, y) →(0,0)

Answer 1

$\lim_{x\to\ 0\\y\to\ 0}f(x,y)= \frac{3\frac{y}{x^3}}{1+2(\frac{y}{x^3})^2}= \frac{3k}{1+2k^2}$ where $k=\frac{y}{x^3}$ .The limit $f(x,y)$ depends on $k$ hence $f(x,y)$ has no limit.