# A bit confused about multivariable limits

I am having a bit of trouble/confusion understanding multivariable limits, specially in regard to approaching by different paths.

Ill explain what I mean;

consider the $$\lim_{(x,y) \to (0,0)}\frac{x^2y}{x^4+y^2}$$

letting $$f(x,y)=\frac{x^2y}{x^4+y^2}$$

we have $$f(x,0)=\frac{0}{x^4}$$ $$f(0,y)=\frac{0}{y^2}$$

so we have then in both cases ( along x and y axis) $f(x,y) \to 0$ as $(x,y) \to (0,0)$

but now consider $$f(x,x^2)=\frac{x^4}{2x^4}=\frac{1}{2}$$ which would approach $1/2$.

So I thought that this showed that the limit does not exist because we obtained different values, however the answer at the back says the limit is 0.

Im sure there is something I must be missing, maybe something very obvious even?

let me know what you guys think, thanks.

• You are right. The limit doesn't exist and you have shown it correctly. The answer at the back is wrong. – mfl Mar 22 '15 at 18:58
• Thank you, should I delete the question? – Quality Mar 22 '15 at 19:09
• You may find some guidance on deletion here or here. To me, the most interesting thing about this question is it shows the value of seeking out a second opinion when your answer disagrees with the book. Sometimes the book is wrong. – David K Mar 22 '15 at 20:12