Usually in the study of limit problems, will encounter more complex problems, in this paper, we discuss how to use the concept of equivalent infinitesimal better limit operation. At the same time, in the process of research, we re-explore the proof of Taylor's formula, and find that some functions have a similar expansion form to Taylor's formula, that is, 'fractional expansion'. It is also found that after the linear combination of Taylor expansion and fractional expansion, the obtained expansion is more accurate, which helps us to have a better understanding of the approximation of function expansion.
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