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quasar987

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## Main Question or Discussion Point

**a definitions for the terms "the limit does not exists"**

Since my textbook doesn't have a definitions for the terms "the limit does not exists" and "the limit goes to infinity", I tried to make them up. I'd like to know if they're correct.

1) Consider [itex] f: \mathcal{D}\longrightarrow \mathbb{R}[/itex] a function and [itex]x_0[/itex] an accumulation point of [itex]\mathcal{D}[/itex]. We say that the limit as x approaches [itex]x_0[/itex] goes to positive infinity (resp. negative infinity) if [itex]\forall M \in \mathbb{R}, \ \exists \delta>0[/itex] such that [itex]x \in \mathcal{D} \cap V'(x_0,\delta) \Rightarrow f(x)>M[/itex] (resp.[itex]f(x)<M[/itex]), and we write

[tex]\lim_{x \rightarrow x_0} f(x) = +\infty \ (\mbox{resp. -\infty})[/tex]

2) Consider [itex] f: \mathcal{D}\longrightarrow \mathbb{R}[/itex] a function. If [itex]\mathcal{D}[/itex] is unbounded superiorly (?) (i.e. has no upper bound), we say that the limit as x approaches positive infinity goes to positive infinity (resp. negative infinity) if [itex]\exists N \in \mathbb{R}[/itex] such that [itex]\forall x \in \mathcal{D}, \ x>N \Rightarrow f(x)[/itex] is strictly increasing (resp. strictly decreasing). In other words, we say that the limit as x approaches positive infinity goes to positive infinity (resp. negative infinity) if [itex]\exists N \in \mathbb{R}[/itex] such that [itex]\forall y, z \in \mathcal{D}[/itex] and [itex] y, z>N, \ z>y \Rightarrow f(z)>f(y)[/itex] (resp.[itex]f(z)<f(y)[/itex]), and we write

[tex]\lim_{x \rightarrow \+\infty} f(x) = +\infty \ (\mbox{resp. -\infty})[/tex]

3) We have an analogous definition for the limit as x goes to negative infinity if the domain has no lower bound. And finally,...

4) Consider [itex] f: \mathcal{D}\longrightarrow \mathbb{R}[/itex] a function and [itex]x_0[/itex] an accumulation point of [itex]\mathcal{D}[/itex]. We say that the limit as x approaches [itex]x_0[/itex], [itex]+\infty[/itex] or [itex]-\infty[/itex] (whichever applies) does not exists if either

i) the limit goes to [itex]+\infty[/itex].

ii) the limit goes to [itex]-\infty[/itex].

iii) the limit is not unique.

Also, if you can think of another definition, or a caracterisation that would make the proofs easier, I'd be very interested to hear it.

Mmh, I can think of one for definition 2 and 3: For 2) "blah, blah" iif for all sequences [itex]\{x_n\}[/itex] such that [itex]x_n \in \mathcal{D}[/itex] and [itex]\{x_n\}[/itex] is strictly increasing for at least all n greater than a certain [itex]N \in \mathbb{R}[/itex] that has [itex]+\infty[/itex] for a limit, the corresponding sequence [itex]\{f(x_n)\}[/itex] has [itex]+\infty[/itex] (resp. [itex]-\infty[/itex]) for a limit. The caracterisation for 3 is analogous.

Phew, this took 45 minutes to write!

Thanks for your replies.

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