limit at infinity proof

**dannyboycurtis** · October 15th 2009, 04:04 PM

Hi I was wondering how I could prove the following:
If a function f is defined on an unbounded above domain $D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\lim_{x\to\infty}f(x) = \infty$
Any help? Thanks

**rn443** · October 15th 2009, 06:41 PM

Originally Posted by dannyboycurtis

Hi I was wondering how I could prove the following:
If a function f is defined on an unbounded above domain $D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\lim_{x\to\infty}f(x) = \infty$
Any help? Thanks

If f is bounded, how could it approach infinity with x?

**tonio** · October 15th 2009, 07:38 PM

Originally Posted by dannyboycurtis

Hi I was wondering how I could prove the following:
If a function f is defined on an unbounded above domain $D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\lim_{x\to\infty}f(x) = \infty$
Any help? Thanks

This is, imo, an odd-worded question: is D, the domain, unbounded above, but then f is "eventually" (what does this mean mathematicalwise?) monotone and bounded? Well, if it has some more or less intuitive meaning, then f is bounded so how can it go to oo??

Try to word out the problem more clearly, perhaps that'll help to undestand what's going in here.

Tonio

**dannyboycurtis** · October 16th 2009, 11:06 AM

Hi I was wondering how I could prove the following:
If a function f is defined on a domain

[WHERE D IS UNBOUNDED ABOVE] and is eventually monotone and eventually bounded,
then

Any help? Thanks

**tonio** · October 16th 2009, 12:56 PM

Originally Posted by dannyboycurtis

Hi I was wondering how I could prove the following:
If a function f is defined on a domain

[WHERE D IS UNBOUNDED ABOVE] and is eventually monotone and eventually bounded,
then

Any help? Thanks

Well, that didn't help, did it? Again the "eventually thing", which is an undefined term in mathematics.
Anyway, IF the function is bounded then it cannot --> oo.

Tonio

**binkypoo** · October 17th 2009, 06:24 PM

If f is defined on a set $D\subseteq\mathbb{R}$ then f is eventually bounded on D if $\exists M>0$ such that $\forall x\in D$ , $|f(x)|\leq M$

If f is defined on a set $D\subseteq\mathbb{R}$ then f is eventually monotone on D if $\forall a,b\in D$ and $a<b$ , either $f(a)\leq f(b)$ or $f(a)\geq f(b)$

I believe if one looks at the function as a sequence the word 'eventually' is definitely defined in mathematics, as in 'eventually' monotone sequences where
$x_{m} > x_{n}$ for some $n,m\geq N\in \mathbb{N}$ but where $x_{m}\leq x_{n}$ for some smaller n and m.

**redsoxfan325** · October 17th 2009, 07:31 PM

Well, those are the definitions of bounded and monotone. The questions everyone's been asking are:

1) What is this "eventually" business?
2) How can anything bounded ever have a limit approaching infinity?

Until some things are cleared up, no one is going to be able to help.

**binkypoo** · October 17th 2009, 07:50 PM

It seems that a sequence a_n can be monotone for n>N but for n<N it may not be. Hence this sequence would be eventually monotone rather than strictly monotone.
like this sequence:
4,3,2,1,2,3,4,5,6,7,8,9,....

**HallsofIvy** · October 18th 2009, 03:40 AM

And we could say that a function, f, is "eventually bounded" if and only if there exist $x_0$ and M such that if $x> x_0$ , then |f(x)|< M.

But we still have the problem that if f is "bounded" or "eventually bounded", its limit cannot be $\infty$ .

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