Question

By considering different paths of approach show that the function f(x,y)=(x^2y)/(x^4+y^2) has no limit as (x,y) (0,0)

Answer 1

Consider a path traced by an arbitrary power function
`y=x^n`, where
`n\in\mathbb Z`. Then


`\displaystyle\lim_((x,y)\to(0,0))(x^2y)/(x^4+y^2)=\lim_(x\to0)(x^(n+2))/(x^4+x^(2n))=\lim_(x\to0)(x^(n-2))/(1+x^(2(n-2)))`

When
`n=2`, we have


`\displaystyle\lim_(x\to0)\frac1{1+1}=\frac12`

but for any larger
`n`, say
`n=3`, we have


`\displaystyle\lim_(x\to0)\frac x{1+x^2}=0`

Therefore the limit does not exist/is path-dependent.