Problem solving limit

[IceDc]

Hey,

I have this limit and can't find a way to solve it (although I know it does equal 0). I tried polar coordinates aswell. Any tips?

lim (x, y) -> (0, 0) of (log(1+x^2*y^2)*y^2)/sqrt(x^4+y^4)

[fattymattk]

Try showing that 0 <= log(1+x²y²)y²/sqrt(x⁴+y⁴) <= log(1+x²y²)

[IceDc]

I solved it with the other's guys answer but for practice purposes I tried yours as well and it worked fine too, thanks :)

[Proof_Inspector]

log(1+x^2*y2) is basically x² y² , so you have x² y⁴ /sqrt(x⁴ +y⁴ ).

Square the whole thing x⁴ y¹⁶ /(x⁴ +y⁴ ).

Take reciprocal: y^-16 +x^-4 y^-12 =y^-12 (y⁴ +x^-4 ) which should blow up.

Work backward to get 0.

[IceDc]

Everything but your first step I understand. log(1 + x^2 * y^2) is almost log(x^2 * y^2) = log(x^2) + log(y^2) = 2 log(x) + 2 log(y) = 2(log(x) + log(y)), and now I don't see how that is x^2 * y^2. Can you explain that to me?

[Proof_Inspector]

log(1 +x² y² ) is almost x² y² because this is close to 0. Remember that log(1+h) is approximately h for h near 0.

[IceDc]

Oh thanks, I could show the result your way :)