In general, the Lagrange multiplier technique is a strategy for finding the extrema of a function subject to equality constraints, see LagrangeMultipliers. One application is the calculation of the critical curves in bivariate limit problems.
To find the critical curves corresponding to the minima and maxima (and saddle points) of a function $f$ we look for the stationary points of $f\left(x\,y\right)$ on the circle parameterized by ${x}^{2}\+{y}^{2}\={r}^{2}$. We formulate the circle condition as $g\left(x\,y\right)\=0$ with $g\left(x\,y\right)\u2254{x}^{2}\+{y}^{2}-{r}^{2}$. The stationary points correspond to those points where the contour lines of$f\left(xcomma;y\right)$ and $g\left(x\,y\right)$ are parallel, i.e. the gradients of the two functions are parallel
$\mathrm{\∇\_\_x,y}f\left(xcomma;y\right)equals;\mathrm{lambda;}\mathrm{nabla;\_\_x,y}g\left(xcomma;y\right)$
with $\mathrm{\λ}$ being the Lagrange multiplier. Equivalently, we can demand that the $\left(2\times 2\right)-$matrix formed by the two vectors $\mathrm{\∇\_\_x,y}f$ and $\mathrm{\∇\_\_x,y}g$ has a vanishing determinant, i.e.
$\mathrm{Det}\left(\genfrac{}{}{0ex}{}{\mathrm{\∂\_\_x}f\mathrm{PartialD;\_\_x}g}{\mathrm{PartialD;\_\_y}f\mathit{}\mathrm{PartialD;\_\_y}g}\right)\equiv \mathrm{PartialD;\_\_x}\mathit{}\mathit{}f\cdot \mathit{}\mathrm{PartialD;\_\_y}\mathit{}g\mathit{-}\mathrm{PartialD;\_\_x}\mathit{}g\mathit{}\mathit{\cdot}\mathit{}\mathrm{PartialD;\_\_y}f\equiv 2y\mathrm{PartialD;\_\_x}\mathit{}\mathit{}f\mathit{-}2\mathit{}x\mathit{}\mathrm{PartialD;\_\_y}fequals;0$. (*)
The solutions of relation (*) correspond to curves in the $\left(x\,y\right)$-plane, e.g. parameterized by $x$ as $y\=y\left(x\right)$. Note that they are independent of the radius $r$. If we were interested in the position of a stationary point along a specific circle, we could insert this curve into the circle condition and obtain the exact points e.g. as $\left(x\,y\left(x\,r\right)\right)$. Instead, here we are interested in the curves themselves since the behavior of $f$ along these curves, when approaching the origin, is crucial in the understanding of the limiting behavior of $f$ as explained above.
Finding all solutions to the constraint equation (*) yields parametrizations for all critical paths. Their limiting behavior proves the existence or non-existence of the bivariate limit. Note that sometimes the relations (*) cannot be solved exactly and approximations are necessary. In the examples below, we will only be interested in the behavior of bivariate functions around $\left(x\,y\right)\=\left(0\,0\right)$. Thus we may expand the solutions into series around this point to simplify the problem when necessary.