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Math Help - Question on notation

  1. #1
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    Question on notation

    Prove \lim_{x \to 0}\frac{1}{x} does not exist.

    Proof:
    Asssume, to the contrary, that \lim_{x \to 0}\frac{1}{x} exists. Then there exists a real number L such that \lim_{x \to 0}\frac{1}{x}=L. Let  \epsilon = 1. Choose an integer n such that n>\lceil 1/\delta \rceil \geq 1. Since n>1/\epsilon, it follows that 0<1/n<\delta. ...........

    Question: In sequence n denotes the term of the sequence. What does n denote in this context?
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  2. #2
    A Plied Mathematician
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    I don't think your proof is quite what's necessary. I think you should show that

    \displaystyle{\lim_{x\to 0^{+}}\frac{1}{x}=\infty.

    Different books sometimes define things differently, but if you can prove that one of the one-sided limits is infinite, then the limit does not exist. So how would you go about proving this? The definition of this limit is as follows:

    \displaystyle{\lim_{x\to a^{+}}f(x)=\infty if and only if

    \displaystyle{(\forall N>0)(\exists\delta>0)(0<x-a<\delta\Rightarrow f(x)>N)}.

    So, you need to prove that

    \displaystyle{{(\forall N>0)(\exists\delta>0)\left(0<x<\delta\Rightarrow \frac{1}{x}>N\right)}.

    In answer to your sequence question, in this context, n is just a number you've defined in your proof.
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