We understand from the infinite related mathematics history that in present classical mathematics (such as classical mathematical analysis and classical set theory) basing on classical infinite theory system, there has been an avoidless theoratical and operational defect whenever we conduct the quantitative cognizings to “infinite related mathematical things” with limit theory--------the confusions of “potential infinite, actual infinite” concepts and the absence of the whole “theory of infinite mathematical carriers” have been making us humans unable to study and cognize scientifically the foundation of limit theory (the “limit theory needs its own foundation” even has never been considered about). And, because of this very fundamental defect, it is very difficult for people to really understand scientifically what kind of mathematical tool limit theory is and how to operate with this mathematical tool scientifically in pracdtical quantitative cognitions to “infinite related mathematical things” in infinite sets. So, following four questions have been produced and troubling people long:

（1）Do we use limit theory treat “potential infinite mathematical things” or “actual infinite mathematical things”？Do we need different limit theories for “potential infinite things” and “actual infinite things”？

（2）Is limit theory a “quantitative cognizing tool” or “qualitative cognizing tool”, a “precise cognizing tool” or “approximate cognizing tool”？

（3）When we conduct the quantitative cognitions to different infinite sets, how can we use limit theory to analyze, manifest and treat those number forms of X--->0 elements inside them (such as those number forms of X--->0 elements in [0, 1] real number set)?

（4）What on earth is the foundation of limit theory？

We understand from the infinite related mathematics history that it is the absence of limit theory’s foundation that results in the production and suspending of so many “infinite related paradox families” in present mathematical analysis and set theory.

Our studies have prooved that limit theory is needed whenever there is the concept of “infinite” in our science and whenever we need to conduct the quantitative cognizings to “infinite related mathematical things”. The emergence of the new infinite theory system (especialy with its “theory of infinite related carriers” and nothing to do at all with “potential infinite--actual infinite”) lays a scientific foundation for limit theory and enable us to answer above four questions clearly and scientifically:

（1）Limit theory has nothing to do at all with “potential infinite mathematical things” or “actual infinite mathematical things”. It is a special mathematical quantitative cognizing tool for “infinite related mathematical carries”. Only one limit theory is needed.

（2）Limit theory is a “quantitative cognizing tool”,------an“approximate quantitative cognizing tool” for “infinite related number forms” in our mathematics (an “1＜1 paradox” was once created to express the nature of limit theory: 1=3×⅓ = 3×0.333333……＜1).

（3）When conducting practical quantitative cognitions with limit theory to mathematical things in infinite sets, what we should do first is “really doing analysis on the infinite related mathematical carriers” being quantitative cognized according to the “theory of mathematical carriers as well as its infinite related number spectrum and set spectrum” in new infinite theory system-------to know what position they are in “new infinite related number spectrum and set spectrum” and what kind of quantitative natures they have, then to decide how to conduct the scientific quantitative cognizing operations with limit theory, but not the indiscriminately “pipeline limit theory operations”.

（4）The new infinite theory system (especially its theory of “infinite related mathematical carriers”) is the foundation of limit theory.

The emergence of the new infinite theory system has decided the emergence of the new limit theory.