What does it mean for a limit to not exist?

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Solution 1

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.

Solution 2

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.

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Updated on August 01, 2022

Comments

  • pinponbong
    pinponbong over 1 year

    I still don't comprehend what people mean by 'limit does not exist', what does it really mean? Ye...

    Thanks.

    • pitchounet
      pitchounet almost 10 years
      For a sequence, for example, it means the sequence is either not bounded or that it is bounded AND you can find two subsequences converging to two different limits.
    • Git Gud
      Git Gud almost 10 years
      Limit of what?${}$
    • pinponbong
      pinponbong almost 10 years
      I mean in general...
    • pinponbong
      pinponbong almost 10 years
      For example what does it mean for some limit as k approaches infinity of some expression A to not exist....
    • Git Gud
      Git Gud almost 10 years
      @pinponbong Expressions don't have limits, functions do.
    • pinponbong
      pinponbong almost 10 years
      Ye, For example what does it mean for some limit as k approaches infinity of some function A(k) to not exist....
    • Admin
      Admin almost 10 years
      This is the negation of: the limit exists
    • Git Gud
      Git Gud almost 10 years
      @pinponbong It means that for all $a\in \mathbb R$, there exists $\varepsilon >0$ such that for all $\delta >0$ and for all $k$ in the domain of $A$, the following statement is true: $$k>\dfrac 1 \varepsilon \land |A(k)-a|\ge\delta.$$
    • rocinante
      rocinante almost 10 years
      I don't think @SamiBenRomdhane designed the question to confuse you more. If you don't really understand what it means for the limit to exist, you're not going to understand what it means for it not to exist.
  • pinponbong
    pinponbong almost 10 years
    I accepted even if I didn't understand the epsilon-delta part so I asked about it here: math.stackexchange.com/questions/627256/….