A definitions for the terms the limit does not exists

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quasar987
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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! :cry:

Thanks for your replies.
 
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  • #2
mathwonk
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1) is correct except that you omitted to say the "limit of what?"

2) looks wrong. i.e it differs greatly from the case in 1) which ti should resemble closely. i.e. the limit of f(x) as x goes to plus infinity, equals plus infinity iff, for every N, there is an M, such that for all x larger than M, f(x) is larger than N.

3) The definition of does nor exist also oooks highly suspicious. In general just negate the rpevious statements.

e.g. the limit of f(x) as x goes to x0 does not exist iff, for all t, there is some e>0, such that for all d > 0, there exists and x closer to x0 than d, and yet with f(x) further from t than e.
 
  • #3
Hurkyl
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IOW, start with the epsilon-delta way to say "the limit exists" and negate it.
 

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