极限思想不仅在高等数学中很重要，在中学数学教学里的体现也非常多。可是一系列的调查表明师生对于极限思想的理解较为浅薄，并且主要应用都是在数学解题方面。事实上极限思想和数系扩张及初等函数之间的联系这两块内容都有着紧密的联系。因此本文针对极限思想的代数方面，首先翻阅人民教育出版社的中小学数学课本，找出其中相关内容，之后借助无穷级数收敛、闭区间套定理、柯西数列收敛、欧拉公式、复变函数相关理论等高等数学知识来证明清楚。叙述内容分为四块：实数理论、初等函数、复数域扩张、复变函数。主要讨论的问题有：无理数的找寻，指数函数与三角函数之间的联系，对数理论的历史发展，代数基本定理，复变函数的定义及联系。本文最后还简要分析了上述讨论的问题在实际教学中的价值和具体策略。

Limit thought is a very important idea, not just in advanced mathematics, but also on primary math teaching. You have to use limit thought if you want to truly understand the development of the number system and the relationship between the elementary function. However, a series of researches show the current situation that many students and teachers just see the surface of the meaning and the applications of limit idea are always on figuring out the math problems. So in this thesis, we first find out the related algebraic contents, then using the advanced mathematics’ knowledge like series convergence, Cantor theorem, Cauchy sequence, Euler’s formula and theorems of complex function to prove them. The contents are divided into four parts: theory of real number, elementary function, the expansion of complex number and complex function. The major issues including: finding the irrational number in the number line, the relationship between exponential function and the trigonometric function, the development of logarithm theory, fundamental theorem of algebra, the definition and relationship of complex functions. In the end, we also simply discuss about the value and specific methods of limit thoughts on practical teaching.