# Thread: limit at infinity proof

1. ## 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

2. 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?

3. 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

4. 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

5. 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

6. 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.

7. 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.

8. 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,....

9. 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 $\displaystyle \infty$.