Corollary:(Divergence criterion for function limits).let f be a function defined on A, and let c be a limit piont of A. If there exist two sequences (x_n) and (y_n) in A with x_n not =c and y_n not =c and lim x_n=lim y_n=c but lim f(x_n) not = lim f(y_n), then we conclude that the functional limit lim f(x) as x->c does not exist.
use this corollary to show that each of the following limits does not exist.
a-lim Absolute value of x/x as x->0.
b-lim g(x) as x->1 where g is Dirichlet's function.
a. Let x_n=1/n and y_n=-1/n. Then x_n->0 and y_n->0 as n->oo. "oo" means infinity. But |x_n|/x_n=1 and |y_n|/y_n=-1. So the limit of |x_n|/x_n is ...
This uses the corollary - divergence criterion for function limits - to show that limits of absolute value and Dirichlet's functions do not exist.