Abstract: The limit of sum form is one of the most important limit problem and has a wide range of applications in all branches of mathematics. It is complex，diverse，and flexible， and there is usually no fixed method to solve it. A postgraduate examination question about the uniform convergence of function sequences triggered the author to think about the limit of sum form with function sequences. By deforming and arranging the limit of the sum form properly and constructing the uniform convergence function，a series of the limit of the sum form with function sequence problems can be solved skillfully. We give rigorous reasoning proof to the conclusion and provide general ideas for solving such problems. Through the analysis of typical examples， it is helpful for beginners to understand and grasp such problems.

Key words: uniform convergence, limit of sum form, limit of function