Give me some hints in calculation this limit. $$\lim_{x\to2\  \\ y\to2}\frac{x^{6}+ \tan (x^{2}-y^{2}) - y^{6}}{\sin(x^{6}-y^{6}) - x^{5}y +xy^{5} + \arctan(x^{2}y -xy^{2})}$$
I used a fact that $$\tan \alpha \sim \alpha \\ \arctan \alpha \sim \alpha \\  \sin\alpha \sim \alpha$$
Since now we have $$\lim_{x\to2\ \\ y\to2}\frac{x^{6}+ x^{2}-y^{2} - y^{6}}{x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}}$$
Then $$\lim_{x\to2\ y\to2}\frac{x^{6}+ x^{2}-y^{2} - y^{6}}{x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}}=\\ \\ 
=\lim_{x\to2\ \\ y\to2}\frac{(x+ y)(x-y)(x^{2}-xy +y^{2})(x^{2}+xy +y^{2})+(x+y)(x-y)}   {(x+ y)(x-y)(x^{2}-xy +y^{2})(x^{2}+xy +y^{2})-xy(x^{2}-y^{2})(x^{2}+y^{2}) - xy(x-y)}=\\=\lim_{x\to2\ \\y\to2}\frac{(x+ y)(x^{2}-xy +y^{2})(x^{2}+xy +y^{2})+(x+y)}   {(x+ y)(x^{2}-xy +y^{2})(x^{2}+xy +y^{2})-xy(x+y)(x^{2}+y^{2}) - xy} $$ 
And what to do next what multipliers to group? Help please.
 A: Observe that
\begin{align*}
\frac{x^{6}+ x^{2}-y^{2} - y^{6}}{x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}}&=\frac{\left(x^{6}+ x^{2}-y^{2} - y^{6}\right)/(x-y)}{\left(x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}\right)/(x-y)}\\
\end{align*}
$$=\frac{x^5+x^4y+x^3y^3+x^2y^3+xy^4+y^5+x+y}{x^5+x^4y+x^3y^3+x^2y^3+xy^4+y^5-xy(x^3+x^2y+xy^2+y^3)+xy}$$
Then
\begin{align*}
\lim_{x\to2\\y\to2}\frac{x^{6}+ x^{2}-y^{2} - y^{6}}{x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}}&=\frac{6\cdot 2^5+2\cdot 2}{6\cdot 2^5-4(4\cdot 2^3)+4}\\
&=\frac{49}{17}
\end{align*}
A: $\lim_\limits{x\to2\ y\to2}\frac{x^{6}+ x^{2}-y^{2} - y^{6}}{x^{6}-y^{6} - x^{5}y +xy^{5} + x^{2}y -xy^{2}}\\
\lim_\limits{x\to2\\\ y\to2}\frac{(x-y)(x^5+x^4y+x^3y^2+x^2y^3+xy^5+y^5) + (x-y)(x+y)}{(x-y)(x^5+x^4y+x^3y2+x^2y^3+xy^5+y^5) - (x-y)(xy)(x^3+x^2y+xy^2+y^3) + xy(x-y)}\\
\lim_\limits{x\to2\\\ y\to2}\frac{(x^5+x^4y+x^3y^2+x^2y^3+xy^5+y^5) + (x+y)}{(x^5+x^4y+x^3y2+x^2y^3+xy^5+y^5) - (xy)(x^3+x^2y+xy^2+y^3) + xy}\\
$
And now let x = y = 2
