# Let f ( x , y ) = ( x 2 </msup> + y 2 </msup> + 2

Let $f\left(x,y\right)=\left({x}^{2}+{y}^{2}+2{\right)}^{xy}$ defined on ${\mathbb{R}}^{2}$ and let $g\left(x,y\right)$ be a degree 2 polynomial in two variables.
Suppose $\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{g\left(x,y\right)-f\left(x,y\right)}{{x}^{2}+{y}^{2}}=0$. Then, determine $g\left(x,y\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

partyjnopp9wa
Recall that for ${e}^{t}=1+t+O\left({t}^{2}\right)$ and $\mathrm{ln}\left(1+t\right)=t+o\left(t\right)$. Therefore,for $\left(x,y\right)\to \left(0,0\right)$
$\begin{array}{rl}\left({x}^{2}+{y}^{2}+2{\right)}^{xy}& =\mathrm{exp}\left(xy\mathrm{ln}\left({x}^{2}+{y}^{2}+2\right)\right)\\ & =1+xy\mathrm{ln}\left({x}^{2}+{y}^{2}+2\right)+O\left({x}^{2}{y}^{2}\right)\\ & =1+xy\left(\mathrm{ln}\left(2\right)+\mathrm{ln}\left(1+\frac{{x}^{2}+{y}^{2}}{2}\right)\right)+o\left({x}^{2}+{y}^{2}\right)\\ & =1+xy\left(\mathrm{ln}\left(2\right)+\frac{{x}^{2}+{y}^{2}}{2}+o\left({x}^{2}+{y}^{2}\right)\right)+o\left({x}^{2}+{y}^{2}\right)\\ & =1+\mathrm{ln}\left(2\right)xy+o\left({x}^{2}+{y}^{2}\right).\end{array}$
Hence $g\left(x,y\right)=1+\mathrm{ln}\left(2\right)xy$. In particular ${a}_{3}=\mathrm{ln}\left(2\right)$