Question

Prove \[\lim_{(x,y) \rightarrow (0,0)} \frac{xy^4}{x^2+y^8}\] does not exist.

Answer 1

approaching along y=x
\[\lim_{(x,x) \rightarrow (0,0)}x^5/(x^2+x^8)=0\]
approaching along y=x^(1/4)
\[\lim_{(x, x^1/4) \rightarrow (0,0)}x^2/2x^2=1/2\neq0\]
therefor, since different paths do not converge on one value the limit does not exist.

Answer 2

Sweet, that was tricky.