The function $f(x,y)=x^y\,,\; (x,y)\in{\mathbb{R}}^2\,,$ is at least twice continuously differentiable in a open disk with centre $(1,1)$ and its 2nd degree Taylor polynomial is \begin{align*} P_{2,f,(1,1)}(x,y)&=f(1,1)+({\rm{grad}}\,{f})(1,1)\cdot(x1,y1)+\frac{1}{2}\,(x1,y1)\,H_f(1,1)\,(x1,y1)^{\top}\\ &=1+(1,0)\cdot(x1,y1)+\frac{1}{2}\,(x1,y1)\,\begin{pmatrix} 0 & 1\\ 1& 0 \end{pmatrix}\,(x1,y1)^{\top}\\ &=1+x1+\frac{1}{2}\,2\,(x1)(y1)\\ &=y\,(x1)+1\,, \end{align*} for which holds \begin{align*} \displaystyle\mathop{\lim}\limits_{(x,y)\to(1,1)}\frac{f(x,y)P_{2,f,(1,1)}(x,y)}{\big\(x1,y1)\big\^2}=0 \quad \Rightarrow\\ \displaystyle\mathop{\lim}\limits_{(x,y)\to(1,1)}{\frac{x^yy(x1)1}{x^2+y^22(x+y1)}}=0\,.\end{align*}
_________________ Grigorios Kostakos
