Try to follow in detales definition of improper limit. Use an example. For instance 

limx=0 1/x.

Can you give more details on what you mean by "demonstrate"?   An object that is infinite in size (i.e., set of natural numbers?)?  Proofs involving infinite sets?  Proofs that use infinite?  Be specific on what you're looking for.

Hope to hear back!

In present traditional infinitude system, the “infinite” can be divided into “potential infinite” and “actual infinite”. What kind of “infinite” do you wand to demonstrate?

The classical demonstration and proof of the uncountability and infinitude of the real numbers is Cantor's diagonal argument, which is included in most introductory texts on set theory.  It was recently discussed (2008) in Nature for its impact on physical theories.  http://www.nature.com/nature/journal/v455/n7215/full/455884a.html

Actually, infinite including infinitesimal and infinity, it is unfair that many people just talk about infinity only but forget infinitesimal which is more difficult to introduce and demonstrate.

It is just because we still know litter about infinitesimals and the infinitesimal related paradoxes have been there troubling us human such as Zeno’s Paradox family.

In addition to the helpful observations made by others in this thread, a good overview approaches to detecting the infinite extent of sets (and sequences) is given in

S. Aztekin, A. Arkin, B. Sriraman, The constructs of PhD students about infinity, The Montana Math. Enthusiast 7(1), 2010, 149-174:

http://www.math.umt.edu/tmme/vol7no1/tmmevol7no1_article9_pp.149_174.pdf

A distinction is made between actual (ordinary Cantor) infinity and potential infinity (p. 150).   A good place to look for a study of infinity (at an elementary level) is

J.A. Petty, The Role of Abstraction in the Conceptualization of Infinity and Infinite Processes, PhD thesis, 1996, Purdue University.

See, also,

J.A. Petty, A comparison of the intuitive foundations of Euclid's and Hilbert's

Geometries, Epistecybernetics 1, 1997, 143-155.

When we discuss our infinite related knowledge since antiquity, intuition was always mentioned. But one thing is important that our science needs ontology and form, we need exact theory and practical operations to deal with real infinite things.

Yes, we don’t reject the role “intuition” plays but it can only offer us inspirations and ideas.

Talking about Cantor's view of infinity, I am a little sorry to say that he cared little for infinitesimal. Does infinitesimal belong to infinite and is infinitesimal anything to do with cardinal numbers or ordinal numbers?

So, that is why I call Cantor's view of infinity “half infinite”; I know many people nowadays will not agree with me, but what I have said is true.