理论上泰勒展开能解决所有极限问题吗？ 第1页

No. Maclaurin series and Taylor series are useful, however, they usually serve for real numbers and does not necessarily deal with potential singularities...

Consider the function , its Maclaurin series is which only converges when , which is not quite applicable. This is because the summation in Taylor series begins from to .

In a Laurent series, the summation begins at , which includes more cases. For example, we can obtain a series expansion for at (the singularity) using Laurent series, which is . This expansion does not seem more useful, however, it is at least more accurate than the Taylor series.

However, your function does not have a Laurent series expansion at . If you indeed want to use series, you must use a Puiseux series. According to the Puiseux's theorem, Puiseux series is the algebraic closure of the field of Laurent series and involves logarithms in the summand and fractional exponents. Like this: . Since grows faster than ,

This is the series expansion of your limit at , where logarithms are involved. In more extreme cases there can be fractional powers, such as: .

I believe that the Puiseux series is the way to solve all elementary limits.