What's the difference between pi and 0.999... ?

According to math professors, the limit of the sequence (.9; .99; .999; ...) is 1 and thus the infinite sum 9/10+9/100+9/1000+... is defined by this limit. In fact, they claim that every real number has a limit. However, no mathematician can tell you what is the limit of pi, e, sqrt(2) or any other irrational number. One difference between pi and 0.999... is that we know the limit of the sequence (0.9; 0.99; 0.999; ...) but we do not know the limit of the sequence (3.1; 3.14; 3.141; ...). So what's the real difference?