Show that if f be a function defined on A, and c be a limit point of A. If there exist two sequences (x_n) and (y_n) in A with x_n not =c y_n not = c and lim x_n=limy_n=c but lim f(x_n) not = f(y_n), then we conclude that the functional limit
lim_x-->c f(x) does not exist.
If the limit of f(x) as x->c exists, suppose the limit is L, then for any e>0, there is a d>0, such that when |x-c|<d, we have ...
A proof involving sequences and limits is provided. The solution is concise.