Question

limit as (x,y)->(0,0) of (xy^2)/(x^2+y^4). problem says to approach from path y=x and x=y^2. I get infinity from both paths but the limit shouldn't exist. any ideas?

Answer 1

use L'hospital's Rule four times when going along y=x\[\lim_{(x,x) \rightarrow (0,0)}x^3/(x^2+x^4)=\lim_{(x,x) \rightarrow (0,0)}0/24=0\]the other is pretty straightforward:\[\lim_{(y^2,y) \rightarrow (0,0)}y^4/2y^4=1/2\]

Answer 2

easier (as apposed to letting y=x) would be to let x=0 and let y->0

the limit would be 0 for this

Answer 3

True, but that was not asked of me.

Answer 4

if you wanted to use y=x ..

\[\lim_{(x,x)\to (0,0)}\frac{x^3}{x^2+x^4}=\lim_{(x,x)\to (0,0)}\frac{x^3}{x^2(1+x^2)}=\lim_{(x,x)\to (0,0)}\frac{x}{1+x^2}=\frac{0}{1}=0\]