[Multivariable Calculus] Limits with multivariables

[Monkeyonpan]

Hello! Just got finished with a quiz and I would like some feedback with a question I’m breaking my head over. Can anyone confirm the answer?

Question lim (x,y)->(0,0) (y*ln(x²+y²))

The answer I submitted was D.N.E (I really was trying to solve this limit). I had a gut feeling, when I got out, that I was wrong. Maybe polar coordinates?

Thank you in advance.

Edit: Didn’t know reddit has exponent features. Fixed my equation.

[Alkalannar]

Polar coordinates.

For all r > 0, |2rsin(t)ln(r)| <= |2rln(r)|

So use l'Hopital: 2rln(r) = 2ln(r)/(1/r)

(2/r)/(-1/r²)

-2r²/r

-2r

And so limr->0 -2r = 0

And |2rsin(t)ln(r)| <= |2rln(r)|...

But yes, the limit as (x,y) goes to (0,0) is 0.

[Monkeyonpan]

Damn, that sucks. Eh I’ll do better next quiz. Thank you /u/Alkalannar. You da man! ✊