Based on a specific case of calculating limit, this paper puts forward an analytical theory for the non convergence caused by the infinite value in this case. This paper proposes that: the algorithm previously applicable on the number field and the complex plane will be extended in the case of infinity, and the algebraic structure of extension can be established. Specifically, the exponential function shows a value approaching 0 under the operation in the case of infinity. The theory can be more c learly explained mathematically through analytical continuation method: firstly, the factorial denominator of Taylor expansion of exponential function that seems to converge on the whole complex plane hides its singular point that it has as the gamma function that has not been continued, and this singularity acts on the number of series items of Taylor expansion and affects the fitting interval of Taylor series. In addition, the analytic continuation function of the exponential function can be found by the method of periodic continuation, so as to demonstrate the feasibility of the analytic continuation mathematically. This paper also makes validation and extended discussion of the theory: by using residue theorem and Jordan lemma to verify the conclusions of the theory, and the residue distribution in Paradise Domain in the direction of pure imaginary number is discussed preliminarily. At last, the paper gives an example of the analysis of divergence product according to the theory, which preliminarily defines an indeterminate product of 0 and infinity and its binary algorithm of multiplication and power, and preliminarily discusses the properties of the two algebraic structures. Finally, it can be obtained that the binary algorithm of multiplication forms a simple and harmonious Abelian semigroup due to the narrow sense of its elements.
