Detecting extrema 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.
Could you be more precise in what you mean? There are many ways to go to infinity in R^2 or C^2
@vdelecroix Sure: what I want is either a limiting value or to know that every value is attainable. I don't care where in the domain I have to go to get there. I want to know if Sage has an algorithmic way to do this. You're right about the ambiguity though, but in practice, I look at single examples at a time and am able to quickly determine limiting behavior. I'd like to do a few hundred (or more) at once though. Does this answer your question?
@vdelecroix P.S. That limiting value can either be a relative extreme or an asymptote. I don't care which.