# Thread: Question on notation

1. ## 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?

2. 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

$\displaystyle{(\forall N>0)(\exists\delta>0)(0N)}.$

So, you need to prove that

$\displaystyle{{(\forall N>0)(\exists\delta>0)\left(0N\right)}.$

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