is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? is there some formal connection?

Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection?

Edit ('13): Perhaps it is more interesting to ask whether limits in category theory can be seen as special limits of ultrafilters or nets.