1, the origin of the concept of limit:
Like all scientific thinking methods, extreme thinking is also the product of abstract thinking in social practice. The idea of limit can be traced back to ancient times. For example, secant technology in Liu Hui is an original and reliable application of the limit idea of "approaching constantly" on the basis of studying intuitive graphics.
The ancient Greeks' exhaustive method also contained the idea of limit, but because of their "fear of infinity", they obviously avoided artificially "taking the limit" and completed the relevant proof by indirect proof-reduction to absurdity.
2. Perfection of the concept of limit;
The perfection of limit thought is closely related to the rigor of calculus. For a long time, many people have tried to solve the problem of the theoretical basis of calculus "satisfactorily", but they have failed. This is because the research object of mathematics has expanded from constant to variable, and people are used to thinking with constant constants.
Classification and nature of limit;
1, classification of limit:
(1) Upper bound: refers to the upper bound of the limit value of the convergent subsequence.
(2) Lower bound: a concept used to judge the lower semicontinuity of a function. Let f(x) be an extended real function defined on point set E. If the lower bound of function f is m(x) in the intersection of δ neighborhood of point X in closure E, the limit of m(x, δ) when δ tends to 0 is called the lower bound function of f(x) along E.
2, the nature of limit:
If a sequence converges, its limit is unique, and for any given positive number ε, there is a positive integer n, so that when n >; When n, the difference between all terms and the limit value is less than ε. In addition, if every term of a sequence is non-negative (or non-positive), then its lower bound (upper bound) is its limit.