The requisite extension assigns to the origin the value of the bivariate limit of $f$ at the origin. Hence, what is required is to show that this limit is 1. Unfortunately, showing $\left|f-1\right|\to 0$ by the "usual" techniques of estimation turns out to be a significant challenge. Instead, a solution based on the algorithm Maple uses for computing bivariate limits is presented. In this algorithm the maxima of $f$ on circles of radius $r$ are computed, and then if it can be shown that these maxima tend to $L$ as $r\to 0$, it will have been established that the bivariate limit of $f$ at the origin is $L$. To this end, express $f$ in polar coordinates and find the extrema for fixed $r$ as a function of $\mathrm{\θ}$.

Since the bivariate limit of $f$ at the origin is 1, the required extension is

$g\left(x\,y\right)\=\{\begin{array}{cc}f\left(x\,y\right)& \left(x\,y\right)\ne \left(0\,0\right)\\ 1& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

Indeed, $\mathrm{limit}\left(f\left(x\,y\right)\,\left\{x\=0\,y\=0\right\}\right)$ = $1$.