limit at infinity proof

• Oct 15th 2009, 04:04 PM
dannyboycurtis
limit at infinity proof
Hi I was wondering how I could prove the following:
If a function f is defined on an unbounded above domain $\displaystyle D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\displaystyle \lim_{x\to\infty}f(x) = \infty$
Any help? Thanks
• Oct 15th 2009, 06:41 PM
rn443
Quote:

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 $\displaystyle D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\displaystyle \lim_{x\to\infty}f(x) = \infty$
Any help? Thanks

If f is bounded, how could it approach infinity with x?
• Oct 15th 2009, 07:38 PM
tonio
Quote:

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 $\displaystyle D\subseteq\mathbb{R}$ and is eventually monotone and eventually bounded,
then $\displaystyle \lim_{x\to\infty}f(x) = \infty$
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
• Oct 16th 2009, 11:06 AM
dannyboycurtis
Hi I was wondering how I could prove the following:
If a function f is defined on a domain http://www.mathhelpforum.com/math-he...dbead2c9-1.gif [WHERE D IS UNBOUNDED ABOVE] and is eventually monotone and eventually bounded,
then http://www.mathhelpforum.com/math-he...43e6ec87-1.gif
Any help? Thanks
• Oct 16th 2009, 12:56 PM
tonio
Quote:

Originally Posted by dannyboycurtis
Hi I was wondering how I could prove the following:
If a function f is defined on a domain http://www.mathhelpforum.com/math-he...dbead2c9-1.gif [WHERE D IS UNBOUNDED ABOVE] and is eventually monotone and eventually bounded,
then http://www.mathhelpforum.com/math-he...43e6ec87-1.gif
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
• Oct 17th 2009, 06:24 PM
binkypoo
If f is defined on a set $\displaystyle D\subseteq\mathbb{R}$ then f is eventually bounded on D if $\displaystyle \exists M>0$ such that $\displaystyle \forall x\in D$, $\displaystyle |f(x)|\leq M$

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

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
$\displaystyle x_{m} > x_{n}$ for some $\displaystyle n,m\geq N\in \mathbb{N}$ but where $\displaystyle x_{m}\leq x_{n}$ for some smaller n and m.
• Oct 17th 2009, 07:31 PM
redsoxfan325
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.
• Oct 17th 2009, 07:50 PM
binkypoo
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,....
• Oct 18th 2009, 03:40 AM
HallsofIvy
And we could say that a function, f, is "eventually bounded" if and only if there exist $\displaystyle x_0$ and M such that if $\displaystyle x> x_0$, then |f(x)|< M.

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