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 (x_{n}) in X converges to a limit x, the sequence (f(x_{n})) 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?