<s>[INST] Problem in showing $\lim_{(x,y)\to (0,0)} \frac{12x^3y^5+4x^4y^4}{x^6+4y^8}=0$ using polar coordinates I'm trying to show that 
$$\lim_{(x,y)\to (0,0)} \frac{12x^3y^5+4x^4y^4}{x^6+4y^8}=0$$
I've used polar coordinates but when I do this I get the possibility of $\frac{0}{0}$  if $\cos(\theta)\to 0$ as $r\to 0$.  So it must be that I need some sort of bound to make sense of this limit.  But I'm not sure how to proceed.
 [/INST] If we wish to use polar coordinates $(r,\theta)$, then we can write
$$\begin{align}
\frac{12x^3y^5+4x^4y^4}{x^6+4y^8}&=r^2\left(\frac{12\cos^3(\theta)\sin^5(\theta)+4\cos^4(\theta)\sin^4(\theta)}{\cos^6(\theta)+4r^2\sin^8(\theta)}\right)\\\\
&=r\left(3\sin(\theta)+\cos(\theta)\right)\left(\frac{4r\cos^3(\theta)\sin^4(\theta)}{\cos^6(\theta)+4r^2\sin^8(\theta)}\right)\tag1
\end{align}$$
Let $ g(r,\theta)=\frac{4r\cos^3(\theta)\sin^4(\theta)}{\cos^6(\theta)+4r^2\sin^8(\theta)}$.  Denote $\sin(\theta)$ by $s$ and $\cos(\theta)$ by $c$.  
We will view $g(r,\theta)$ as a function of $\theta\in \mathbb{R}$, which is differentiable and $2\pi$-periodic.  Therefore the extrema occur at points for which $\frac{\partial g(r,\theta)}{\partial \theta}=0$.  
Then, taking the partial derivative with respect to $\theta$, and , we have 
$$\begin{align}
\frac{\partial g(r,\theta)}{\partial \theta}&=4rs^3c^2\,\left(\frac{(4c^2-3s^2)(c^6+4r^2s^8)-s^2c^2(32r^2s^6-6c^4)}{(c^6+4r^2s^8)^2}\right)\\\\
&= 4rs^3c^2\,\left(\frac{(3+c^2)(c^6-4r^2s^8)}{(c^6+4r^2s^8)^2}\right)\tag 2
\end{align}$$
We see that $\frac{\partial g(r,\theta)}{\partial \theta}=0$ when $\sin(\theta)=0$ or $\cos(\theta)=0$ or $\cos^6(\theta)=4r^2\sin^8(\theta)$.  
When $\sin(\theta)=0$ or $\cos(\theta)=0$, $g(r,\theta)=0$.  When $\cos(\theta)^6=4r^2\sin^8(\theta)$, 
$$g(r,\theta)=\text{sgn}(\cos(\theta))\tag 3$$
Finally, using $(3)$ in $(1)$ reveals
$$\left|\frac{12x^3y^5+4x^4y^4}{x^6+4y^8}\right|\le r|3\sin(\theta)+\cos(\theta)|$$
whereupon applying the squeeze theorem yields the coveted limit

$$\bbox[5px,border:2px solid #C0A000]{\lim_{(x,y)\to (0,0)}\frac{12x^3y^5+4x^4y^4}{x^6+4y^8}=0}$$

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