<s>[INST] The limit of $\frac{x^2 + y^2 - x^{3}y^{3}}{ x^2 + y^2}$ as $(x,y)\to(0,0)$ I have formula: $$ f(x,y) = \frac{x^2 + y^2 - x^{3}y^{3}}{ x^2 + y^2}$$ 
and I have to find whether $ \lim_{(x,y) \rightarrow (0,0) }  f(x,y)$ exists .
My attempt was: 
(1) set $x=0$; then $f(0,y) = 1$, set $y=0$, then $f(x,0) =1$.
I concluded that this function probably has $L = 1$. 
By applying the definition of limit in two variables, 
 if $0 < \sqrt{x^2+y^2 }< \delta $, then there exists $|(-xy)^3 + x^2 + y^2) / 2 - 1 |  < \beta$. 
I simplified all the way up to 
$(|(xy)^3|/(x^2+y^2) ) < \beta $, then I hit the brick wall.
It appears that $(xy)^3$ is unbounded, and I can't simplify this for the life of me. What am I doing wrong? 
 [/INST] Use polar coordinates: $x=r\cos(\theta), y= r\sin(\theta), 0\leq \theta < 2\pi$.
Then we get, for $(x,y)\neq(0,0)$, that $f(x,y)=\frac{x^2+y^2-(xy)^3}{x^2+y^2}=1-\frac{(xy)^3}{x^2+y^2}=1-\frac{(r^2\cos(\theta)\sin(\theta))^3}{r^2}=1-r^4(\cos(\theta)\sin(\theta))^3\rightarrow 1$ when $r\rightarrow 0, \theta$ varies freely.
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