Give me some hints in calculation this limit.

Question

$$\lim_{x\to2\ \\ y\to2}\frac{x^{6}+ \tan (x^{2}-y^{2}) - y^{6}}{\sin(x^{6}-y^{6}) - x^{5}y +xy^{5} + \arctan(x^{2}y -xy^{2})}$$ 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.

Answer 1

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*}

Answer 2

$\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