Your notation is confusing. Let's say you are using . Then . Now, . Is that what you mean? They both converge, but not to the same limit. That is proof that the limit does not exist.
So the limit as a sequence is defined as limn→∞ an = L , or more precisely as if for all ǫ > 0 there existsN ∈ N so that for all n ≥ N,
|an − L| < ǫ.
And then the sequential notion of continuity is defined as: for some functionf: X → Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x).
So can this idea be used to disprove a limit like as x->0, cos(1/x) -> no limit .
For instance, if one supposes that two sequences x_1 = 1/(2npi) and some other sequence x_2 both tend to 0, as n->infinity, then can one state
if f(x)=cos(1/x) then f(x_1)=1 and f(x_2) = 0 (suppose this is the case), which indicates that the limit does not exist.
My understanding is that since both sequences tend to 0, we can use x_1 and x_2 as x in the function f by the above defn. And since f(x_1) and f(x_2) don't converge, the limit does not exist.
Please correct me if my reasoning is wrong?