The solution to the limit is as follows: (I have listed everything that can be listed! ! ! ! ! Do you have anything to add? )
1 is equivalent to infinitesimal transformation (it can only be used in multiplication and division, but it doesn't mean that it can't be used in addition and subtraction, just to prove that the limit still exists after division). The x power of e-1 or the a power of (1+x)-1 is equivalent to Ax and so on. Study hard.
(When X approaches infinity, it reverts to infinitesimal.)
2 the rule of writing (big topics sometimes imply that you want to use this method) First of all, there are strict preconditions for its use! ! ! ! ! ! It must be x close, not n close! ! ! ! ! ! ! (So, when facing the limit of sequence, we must first convert it into the limit in the case of seeking X approximation. Of course, N approximation is only a case of X approximation, and it is a necessary condition.
(One more thing. Of course, the limit n of the sequence is close to positive infinity, and it can't be negative infinity! ) must be the derivative of the function to exist! ! ! ! ! ! ! ! (If I tell you g(x), but I don't tell you whether it is differentiable, it will undoubtedly be directly used to die! ! )
Must be 0 to 0, infinity to infinity! ! ! ! ! ! ! ! ! Of course, we should also pay attention to the situation that the denominator cannot be 0 and other rules are divided by 3.
1 0 to 0 Infinite than Infinite can be used directly.
2 0 times infinity MINUS infinity (the relationship between infinity and infinitesimal is reciprocal), so infinity is written as the reciprocal form of infinitesimal. After the general term, it can be changed into the form of 1, which is the 0 th power of 3, the infinite power of 1, and the infinite 0 th power.
For the (exponential power) equation, the main method is to take exponent and logarithm, so that the function on the power moves down and is written in the form of 0 and infinity. That's why there are only three forms. When the two ends of LNx approach infinity, its power moves down and approaches 0. When his power moves down and approaches infinity, LNX approaches 0. )
3 Taylor formula (when it contains the x power of e, especially the addition and subtraction of positive co-operation, we should pay special attention! ! ! ! ) x expansion e Sina expansion cos expansion ln 1+x expansion is very helpful to simplify the topic.
For the solution of infinite form on infinity, take the big head principle and divide the largest term by the numerator denominator! ! ! ! ! ! ! ! ! ! ! It looks complicated and easy to handle! ! ! ! ! ! ! ! ! !
5 infinitely less than bounded function processing method
We must pay attention to this method when the complex variable function, especially the positive coincidence complex variable function is multiplied with other functions.
Faced with a very complex function, you may only need to know its scope, and the result will come out! ! !
6 pinch theorem (mainly dealing with the limit of sequence! )
This is mainly to see that the function in the limit is in the form of equation division, scaling and expansion.
7 Application of proportional arithmetic progression formula (dealing with the limit of sequence) (symbol with absolute value of q less than 1)
8 The splitting and adding of items (eliminating the majority in the middle) (dealing with the limit of series) can be decomposed into simplified functions by undetermined coefficient method.
9 How to find the left and right limits (dealing with the limit of sequence) For example, if we know the relationship between Xn and xn+ 1, then the limit of Xn is the same as that of Xn+ 1, and the limit value of the finite term should be removed for the limit.
The application of 100 two important limits. These two are very important! ! ! ! ! For the first one, it is the ratio of sinx to x when x approaches 0. If x approaches infinity, there are two pairs of infinitesimals (these two pairs of infinitesimals are actually infinite forms for the function 1) (when the cardinality is 1, two important limits should be paid special attention to).
1 1 There is another method, which is very convenient when approaching infinity.
Different functions approach infinity at different speeds! ! ! ! ! ! ! ! ! ! ! ! ! ! !
The x power of x is faster than x! Faster than exponential function, faster than exponential function, faster than logarithmic function (you can also see the drawing speed)! ! ! ! ! !
When x approaches infinity, the limit of their ratio becomes clear at a glance.
12 method of substitution is a skill. A topic doesn't have to exchange elements, but exchange elements will be mixed.
13 if you want to calculate, the four algorithms are also a method, and of course they are mixed.
There is another way to deal with the limit of 14 sequence.
That is, when you are really cornered in the face of a problem, you can consider converting it into a definite integral. Generally from 0 to 1.
Monotonicity and Boundedness of 15
Prove monotonicity when dealing with recursive sequences! ! ! ! ! !
16 directly uses the definition of derivative to find the limit.
(Generally, when X approaches 0, f(x plus or minus one value) is plus or minus f(x) on the molecule, so I pay special attention to it. )
Hope to adopt, thank you.