### Detecting extreme and asymptotes of (nasty) functions of two variables

I have a rational function of two variables* whose extrema and limiting behavior I am intersted in. Basically I want to know what values it *cannot* contain, so, I want to know limits and asymptotics. I could do this more or less manually by plotting at zooming in, zooming out until I come up with a thesis, and then taking limits to verify, but I want to do a large number of examples at once, and I'd like to write a function that just spits out the relevant values without having me look at the plot. Any ideas? A quick search turned up no useful information on taking 2D limits.

*For those interested, the function is the j-invariant of an elliptic curve, and I'm interested in a family of such curves. It's a function of two variables as the curve is found as a particular hyperplane section of a surface, and I have a two parameter family.