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.