In order to understand the behavior of a bivariate function $f$ when approaching a point in ${\mathrm{ℝ}}^2$, let us first look at its behavior in a small neighborhood around this point. In particular, we are interested in the minima and maxima of the function in this region and their behavior as we approach the limit point. Let us illustrate this by looking at the function

$\mathrm{f\_\_}\left(x\,y\right)\=\frac{{\left(x\+y-{\left(x-y\right)}^2\right)}^2}{x^2\+y^2}$

and by studying its behavior for $\left(x\,y\right) \to \left(0\,0\right)$. We start by examining $f$ on small circles in the $\left(x\,y\right)$-plane about the origin. Their radius shall be denoted by $r$ and we will later let $r \to 0$.

Fig.1 shows a contour plot of $\mathrm{f\_\_}$. Hover over the plot to see its values in the different regions. Furthermore, a circle is drawn about the origin. The solid circles and squares indicate the positions of minima and maxima, respectively, of $\mathrm{f\_\_}$ along the circle. Using the slider you can increase and decrease the radius of the circle and thus track the position of these points. By checking the "Show curves of minima and maxima" checkbox the plot displays the paths connecting these minima and maxima. Clicking on "Play" will animate the process $r \to 0.$

If the function $\mathrm{f\_\_}$ approaches the same value for $r \to 0$ along these paths, then all possible paths will approach this value since they cannot give a larger or smaller value than that of the maximum or minimum, respectively. If they instead give different limiting values, then the bivariate limit does not exist and the procedure returns an upper and lower bound of the values of $f$ at that point. Thus, finding the paths that connect minima and maxima of a function near the origin reduces the calculation of its bivariate limit to a finite number of one-dimensional limits along these paths which can be calculated by the standard means for one-dimensional limits.

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 $g\left(x\,y\right)≔x^2\+y^2-r^2$. The stationary points correspond to those points where the contour lines of$f\left(x\,y\right)$ and $g\left(x\,y\right)$ are parallel, i.e. the gradients of the two functions are parallel        

$\mathrm{\∇\_\_x,y} f \left(x\,y\right)\=\mathrm{\λ} \mathrm{\∇\_\_x,y} g\left(x\,y\right)$

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.

  $\mathrm{Det}\left(\genfrac{}{}{0ex}{}{\mathrm{\∂\_\_x} f \mathrm{\∂\_\_x} g}{\mathrm{\∂\_\_y} f \mathit{ }\mathrm{\∂\_\_y} g}\right)\equiv \mathrm{\∂\_\_x}\mathit{ }\mathit{ }f \cdot \mathit{ }\mathrm{\∂\_\_y}\mathit{ }g\mathit{-}\mathrm{\∂\_\_x}\mathit{ }g\mathit{ }\mathit{\cdot }\mathit{ }\mathrm{\∂\_\_y} f \equiv 2 y \mathrm{\∂\_\_x}\mathit{ }\mathit{ }f \mathit{-}2\mathit{ }x\mathit{ }\mathrm{\∂\_\_y} f \=0$.            (*)

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.

In Maple you can calculate the bivariate limit of real rational functions via the usual limit command and its multidimensional extension limit/multi. It calculates the limit via the technique explained above: using the Lagrange multiplier technique to find critical paths and calculating the one-dimensional limits along these curves. So one can calculate the bivariate limit of the example above via

Thus the bivariate limit of this function does not exist, but the function approaches values in the interval 0..2 at the origin depending on the path. You may verify this result visually by plotting the function as a three-dimensional surface plot:

Using the eval command you can also calculate the one-dimensional limit along a specific curve:

Further details, like the explicit form of the constraint equation, the critical curves and their one-dimensional limits, are given in the example $\mathrm{f\_\_14}$ below. For further information on the implementation of the bivariate limit command, please visit the Maple help pages BivariateLimits2017, BivariateLimits17 and BivariateLimits2015.