Question

Let #f(x,y)={(x²y)/(x⁴+y²), " if " x⁴+y²≠0, {0, " if " x=y=0#. Check whether `lim_((x,y)→(0,0))f(x,y)` exists or not?

Answer 1

See below.

Explanation:

Choosing a line such that `y = lambda x` we have

`lim_((x,y)->(0,0))f(x,y) = lim_(x->0)(x^3lambda)/(lambda^2 x^2 + x^4) = lim_(x->0)(xlambda)/(lambda^2 + x^2)=0`

but now choosing a curve such that `y = x^2` we have

`lim_((x,y)->(0,0))f(x,y) = lim_(x->0)x^4/(x^4+x^4) = 1/2` then

`lim_((x,y)->(0,0))f(x,y)` does not exists.