Does $\lim_{(x,y)\to(0,0)}\frac{x^2+y^4}{x+y^2}$ exist, and equal $0$?

I just need a simple yes or no answer to see if my answer is in line with a consensus. Does the limit $$\lim_{(x,y)\to(0,0)}\frac{x^2+y^4}{x+y^2}$$ exist? I think it does, and the limit is 0.

-
"I think it does, and the limit is $0$". Can you write down your reasoning? –  Jack Apr 9 '12 at 4:49
add comment

1 Answer

Hint: Note the denominator is equal to zero along the parabola $x = y^2$. That's going to make it impossible for the limit to exist.

-
Ah, I think I understand it better if I think of the limit in terms of polar coordinates. There world be a cos(theta) remaining in the denominator, and since that varies, the limit could not possibly exist. –  shmiggens Apr 9 '12 at 4:54
@shmiggens: The $\cos\theta$ in the denominator can be viewed as relevant, but be careful not to apply that reasoning to say $\dfrac{x^6+y^8}{x^2+y^4}$. –  André Nicolas Apr 9 '12 at 5:50
add comment