Indetifying limit in two variable function is really much harder than one variable function case. We have to check every possible approaching direction to limiting point. In one variable function, domain is a subset of real line number. So, we have only two possibilities while approaching to the point x=a. On the other hand, in two variable function domain is a subset of two dimensional plane. So, endless possibilities exist while approaching to the point (x, y) =(a, b). In the following figure, only four of these possibilities has been shown.
If at least two of the approaching path gives different limit values, we can conclude that limit does not exist.
On the other hand, it is very hard to identify if limit exist, particularly at discontinuity point.
It is believed that polar coordinate transformation is very helpfull to identify the limit at even discontinuity point. Is it really correct for every cases?
Let our example be f(x,y) = (x^3*y)/(x^6+y^2) and try to explore limit at (x, y) = (0,0) if exists.
To calculate this limit, our starting point will be writing the function in polar coordinates. Then we will assume that the limit of function in polar coordinate while r approaches to 0 will simulate all paths approaching to (0,0). So our limit would be transformed into a one variable limit. Let's see if it does actually work.
At last, we found out that the limit is 0. I have also checked it by a popular online limit calculator platform. It have also given the same result.
After this algebraic verification from an outer source, I was almost convinced. Then, I decided to observe the function's graph by using a 3d plotter. I used GeoGebra (https://www.geogebra.org/3d)
When I observe the behavior of surface at (0, 0) it was not definitely convincing. Even if it seems like the limit is 0, something is wrong. Because insufficient resolution may hide som facts. Besides, when I zoom in to (0, 0) the appearance did not change. Then, I decided to look for some other paths which may not give same limit value.
What does polar coordinates actually simulate?
The strategy of changing cartesian coordinates into polar coordinates give us the opportunity of checking all linear approaching paths to the limiting point at the same time. You may ask "why do we need polar coordinates for that? Could we not use the lines y=mx to approach (0,0)". The answer is simply "No!". Because, the paths y=mx can not represent the line x=0, while polar coordinates could represent the path x=0 for the angle's 90 degrees value.
In the calculation above, we found that all linear paths approaching to (0,0) give same limit value. But some curvilinear have risk of different limit value.
In our case, assume that we are approaching to (0,0) on the line y=x and on the curve y=x^3. and see what is happenning;
This result is explaning everything. Since the limit while approaching to (0, 0) on the path y=x^3 is 1/2 while linear paths approaching to (0,0) gives the limit 0, limit does not exists.
What learned from this exceptional case?
It is not safe to conclude that "limit exists" by using polar coordinate strategy as transforming to polar coordinates, then just calculating the limit similar to one variable functions. Because we have still a two variable function having r and theta as two independent variables.