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My freshman year. Our senior math teacher said that the infinitesimal substitution rule is not suitable for addition and subtraction, but I see that some problems also use the substitution rule? How do you understand the teacher's words? Under what circumstances does the infinitesimal substitution rule not apply? Can you give examples, thank you! If you are satisfied, there is an addition
The Equivalent Infinitesimal Substitution can be used in multiplication and division. Sometimes it can be used in addition and subtraction, sometimes it can't be used. For the course of advanced mathematics, it's enough to remember that addition and subtraction can't be used. Generally, it's not necessary to use the Equivalent Infinitesimal Substitution in addition and subtraction. I believe most of the substitutions in addition and subtraction you see are not equivalent infinitesimal substitutions, but four arithmetic operations of limit
For example: Lim [x → 0] (1 + cosx) (SiNx) / x = Lim [x → 0] 2 (SiNx) / x = 2, here are actually four algorithms
lim[x→0] (1+cosx)(sinx)/x=lim[x→0] (1+cosx)lim[x→0] (sinx)/x
Equivalent Infinitesimal Substitution is used in addition and subtraction, such as the following question:
Example 1: Lim [x → 0] (SiNx + x) / TaNx = Lim [x → 0] (x + x) / x = 2
But sometimes the use of addition and subtraction is not good, such as
Example 2: Lim [x → 0] (x-sinx) / X & #, if we change SiNx of molecule to x, it is wrong
Why can it be changed sometimes? Why can't it be changed sometimes? This problem can only be explained clearly by using Taylor formula. I wonder if you have learned Taylor formula. I won't go into details first. I'll briefly talk about a principle. If you need a more detailed explanation, you can ask me
Principle: if the substituted item is deleted in the addition and subtraction operation, then the substitution cannot be done. If it is not deleted, it can be done. For example 1, after SiNx is replaced by X, X does not disappear, so the substitution is OK. In example 2, after SiNx is replaced by x, the subtraction from the previous x just disappears, and the substitution is not OK
I hope I can help you. If you don't understand, you can ask. If you solve the problem, please click the "select as satisfactory answer" button below. Thank you
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