# Thread: Sequential Characterization of limit

1. ## Sequential Characterization of limit

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 function
f: 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?

2. ## Re: Sequential Characterization of limit

Your notation is confusing. Let's say you are using $a_n = \dfrac{1}{2n\pi}, b_n = \dfrac{2}{(2n-1)\pi}$. Then $\lim_{n\to \infty}a_n = \lim_{n\to\infty}b_n = \lim_{x\to 0}x = 0$. Now, $\lim_{n\to \infty} \cos\left(\dfrac{1}{a_n}\right) = 1 \neq 0 = \lim_{n\to \infty} \cos\left(\dfrac{1}{b_n}\right)$. Is that what you mean? They both converge, but not to the same limit. That is proof that the limit $\lim_{x \to 0} \cos\left(\dfrac{1}{x}\right)$ does not exist.

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### characterising the seqences for the limit tending to infinity

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