Improving chances of selection into a course at university
Limited admission courses can happen at any level, for a number of reasons.  In my own experience, it has tended to be some combination of three cases:


The department intends the limited admission to be a filter, to redirect "unpromising" students elsewhere.
Resources are simply limited, and only a certain number of students can be supported (this is especially the case for courses that are experiments with the curriculum).
"Permission of instructor needed": this is some sort of special or favored upper level course, and the instructor is aiming for something more like a seminar or reading group than an ordinary course.


If you're dealing with the first case, then there may be a formal standard, and you're just out of luck.  Much of the time, however, professors have some leeway in how they handle selection.  In some cases, students are actually encouraged to present their case (but if that was the case, you would know).  As an instructor, I would rather any day be dealing with an enthusiastic and hardworking student than one who's just there because it's part of their program, even if the hardworking student is somewhat more poorly rated.  

So yes, by all means get in contact, express your enthusiasm and background, and see what the instructor says.  If you're polite and willing to take no for an answer, then the worst that can happen is they say no.