I still don't comprehend what people mean by 'limit does not exist', what does it really mean? Ye...
Thanks.
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$\begingroup$Morally, the limit $$\lim_{k \to \infty} A(k)$$ does not exist if the values $A(k)$ don't settle down on a specific number as $k$ grows. For example, $\frac{1}{k}$ settles down to zero, while $\sin(k)$ just oscillates back and forth.
Formally, we say the limit of a sequence $A(k)$ is $L$ if and only if for every error $\epsilon$, you can find an $N$ large enough such that beyond $N$, $A$ is within $\epsilon$-error of $L$. That is, for all $n \ge N$, $|A(n) - L| < \epsilon$.
The limit fails to be $L$ when the negation of the statement is true: That is, there is some error $\epsilon$ such that for arbitrarily large $k$, we have $|A_k - L| \ge \epsilon$. If this happens for all real $L$, then the limit doesn't exist.
$\endgroup$ 1 $\begingroup$Let $(a_n)_{n=1}^\infty$ be a sequence (e.g. of real numbers). We say that the limit of the sequence exists if there exists a real number $a$ such that for arbitrary $\epsilon>0$ there exists $n\in\mathbb N$ such that $|a_m-a|<\epsilon$ for all $m>n$.
If this is not the case, that is if for all real numbers $a$ there exists some $\epsilon>0$ such that for all $n\in\mathbb N$ there exists $m>n$ with $|a_m-a|\ge \epsilon$, then we say that the limit does not exist.
$\endgroup$ $\begingroup$eg : $$\lim_{x\to\infty} \sin(x) = ?$$
$\endgroup$ $\begingroup$I am going to depart from the delta-epsilon definition and treat this in softer terms because nobody really thinks in terms of deltas and epsilons at first.
What does it mean for a limit to exist, e.g. lim f(x) = 1? It means that as x gets larger and larger, the value of the function gets closer and closer to 1.
If the limit does not exist, this is not true. In other words, as the value of x increases, function value f(x) does not get close and closer to 1 (or any other number). You can't say anything about what the value of f(x) will be as x increases.
$\endgroup$ $\begingroup$As you say "limit does net exist" sounds more like one would expect that there should be a limit but its simply missin, like in the case of an incomplete metric space. So imagine a sequence net filter etc being cauchy but theres simply the limit missin, e.g. X=R+{0} and x_n=1/n.
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