Mathematics equivalent infinitesimal limit

Answer 1

When using the equivalent infinitesimal to find the mathematical limit, there were the following points:
1. ** Conditions **
- When the numerator or decimal in the relation is the product or division of a limited term, if the limit of a certain part exists and is not zero, it can be calculated first, and the equivalent infinitesimal can be used to replace it.
- When the numerator or decimal in the relation was in the form of addition and substitution, the equivalent infinitesimal could only be applied when the accuracy was achieved.
2. ** Fusion with other methods **
- It was often used together with the L'Empida's Law. There were three conditions for the application of L'Empida's Law, and it was even easier to combine it with the equivalent infinitesimal. For example, when finding the limits of some functions, the first of the two key limits could be calculated using the equivalent infinitesimal, but one had to master the inequation used in the proof process. The second key limit should not be too rigid when applied. When finding the limit of a power-exponential function, one could first see if it could be converted to the structure of a exponential function, and then calculate the limit and simplify it with the equivalent infinitesimal.
3. ** The equivalent infinitesimal relation of common functions **
- You can use the McLaughlin formula for common functions (with the Peiano remainder) to determine the equivalent infinitesimal relationship. However, it should be noted that the equivalent infinitesimal is expanded at x = 0, so it can only be applied to the case of x = 0. Since the first few terms of Taylor's expansion are only approximate, one or two basic questions are enough. If it is a difficult question, you have to analyze the specific problem and take a few.
4. ** Special circumstances **
- For zero-to-zero problems, the L'Bizda rule might be used.(The "low-n-one" in the L'Bizda rule mentioned here is not standard, but it is actually the L'Bizda rule, which is the operation of deriving the numerator and Denominator respectively.) When there are two scores in a question, it can be changed into a score line through the general score arrangement to continue the calculation.
- If you encounter the limit of the infinite power of 1, you can transform it with the log identity. Add one and subtract one in the log identity, which is equivalent to u - 1.
- When finding the limit of a sequence of numbers, if the limit of a sequence of numbers was a question of "the sum of infinitely many infinitesimal terms, and the form of each term is very standard", one could consider whether the limit could be regarded as a special definition of the definite integral of a function. It may be difficult to directly use the integral definition of some related limit of a sequence of numbers. In this case, it was necessary to use the convergence criterion and other methods to discuss. It could also be used to find the limit of a sequence of numbers. It played an important role in finding the limit of a sequence of numbers, often making the limit of a sequence of numbers easier and faster to solve. In addition, when finding the limit of the sequence of numbers, you can use the pinching criterion, that is, first find two sequences, such that the sequence of numbers exists and is equal to the sequence of numbers. When the above methods could not find the limit, one could also try Taylor's formula, but one had to remember the nine McLaughlin expansions.
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