I have been trying to find the limit $$\lim_{(x,y)\to (0,0)} \frac{e^{(x+y)^2}-\cos(x^2+y^2)-2xy}{x^2+y^2-\sin(x-y)^2+2xy}$$ using taylor series and little-o notation for the "remainder", and am stuck on where to proceed. Did I make some arithmetic error or am I missing something big here? Thanks!
Observe that: $$e^{(x+y)^2} = 1+(x+y)^2+o(x+y)^2$$, $$\cos(x^2+y^2) = 1+o(x^2+y^2)$$, and $$\sin^2(x-y) = \left(x-y +o(x-y)^2\right)^2 = (x-y)^2 + o(x-y)$$
With this, I got:
$$\lim_{(x,y)\to (0,0)} \dfrac{x^2+y^2+o(x+y)^2+o(x^2+y^2)}{4xy+o(x-y)} = \lim_{(x,y)\to (0,0)} \dfrac{x^2+y^2+o(2x^2+xy+y^2)}{4xy+o(x-y)}$$
I don't really feel too confident in this result, and I'm not really sure where to go from here. Thanks for the help!