Let |Xn| be defined as an infinite sequence. If there is a constant a for any given positive number ε (no matter how small it is), there is always a positive integer n, so that when n >; All Xn when n has inequality | xn-a |
Lim Xn = a or Xn→a(n→∞)
If the sequence has no limit, it is said that the sequence diverges.
nature
1. Uniqueness: If the limit of the sequence exists, the limit value is unique, and the limit of its subsequence is equal to the limit of the original sequence;
2. Boundedness: If a sequence {xn} converges (has a limit), then this sequence {xn} must be bounded.
However, if a series is bounded, it may not converge. For example, {xn}: 1,-1,-1, ... (-1) n+ 1, ...
3. number preservation: if a sequence {xn} converges to a, and a >；; 0 (or a
4. The relationship between convergent sequences and other subsequences: (Generally speaking, changing the finite term of a sequence does not change the limit of the sequence. If the sequence {xn} converges to a, then any of its subsequences also converges, and the limit is also a.
When n→∞ is the limit of commonly used sequence, there are
An=c the limit is C.
An= 1/n is limited to 0.
The limit of An = x n (∣ x ∣ less than 1) is 0.
Sufficient conditions for the existence of sequence limit
Pinch principle has series {An}, {Bn} and {Cn}, which satisfy an ≤ bn ≤ cn and n ∈ z *. If lim An = lim Cn = a,
Then there is limbn = a.
Monotone convergence theorem monotone bounded sequence must converge. [is one of the important conclusions of real number system, and its important application is to prove the existence of limit lim (1+ 1/n) n]
Cauchy convergence criterion let {Xn} be a sequence, if ε >; 0, there is N∈Z*, as long as n satisfies N >；; N, then for any positive integer p, there is
| X(n+p)-Xn | & lt； ε. Such a series {Xn} is called Cauchy series.
This asymptotic stability is equivalent to convergence. That is, they are necessary and sufficient conditions.
functional limit
The professionally defined set function f(x) has a definition in the centripetal neighborhood at point X. If there is a constant A, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies inequality 0,
| f(x)-A | & lt； ε
Then the constant a is called the time limit of the function f(x) when x → x
The popular definition is 1. Let the function y=f(x) be defined in (a, +∞). If the function f(x) is infinitely close to a constant a when x→+∞, then A is called the limit of the function f(x) when x tends to +∞. Let limf(x)=A, x→+∞. 2. Let the function y=f(x) be defined near point A. When x approaches a infinitely (denoted as x→a), the value of the function approaches a constant infinitely, then A is called the limit of the function f(x) when x approaches a infinitely. Write lim f(x)=A, x → a.
Left and right limits of function 1: If x approaches x0 from the left side of point x=x0 (that is, x < x0) and function f(x) infinitely approaches constant a, then a is called the left limit of function f(x) at point x0, and it is denoted as x→ x0-LIMF (x) = a. 。
2. if x is from the right side of point x=x0 (i.e. x >；; When x0) infinitely approaches the point x0, the function f(x) infinitely approaches the constant a, that is, A is the right limit of the function f(x) at the point x0, which is denoted as x→ x0+LIMF (x) = a. 。
Note: If the left and right limits of a function are different on x(0), then the function has no limit on x(0).
Whether a function has a limit at x(0) has nothing to do with whether it is defined at x=x(0), as long as y=f(x) is defined near x(0).
Two important limits: 1, x→0, sin(x)/x → 1.
2.x→0, (1+X) 1/X→ E or x→∞, (1+1/x) x→ e.
x→∞,( 1+ 1/x)^( 1/x)→ 1
(where e≈2.7 1828 18 ... is an irrational number)
Function limit algorithm
Let lim f(x) and lim g(x) exist, let lim f (x) = a, and lim g (x) = b, then there are the following algorithms.
Linear operation addition and subtraction:
lim ( f(x) g(x) )= A B
Digital multiplication:
Lim( c* f(x))= c * A (where c is a constant)
Nonlinear operation multiplication and division;
lim( f(x) * g(x))= A * B
Lim( f(x)/g(x)) = A/B (where B≠0)
Power:
Forest (f (x)) n = n
The concept of limit
An introduction to limit thought is an important thought in modern mathematics, and mathematical analysis is a subject that studies functions based on the concept of limit and with limit theory (including series) as the main tool.
The so-called limit thought refers to a mathematical thought that uses the concept of limit to analyze and solve problems. The general steps to solve problems with extreme ideas can be summarized as follows:
For the unknown quantity investigated, first try to conceive a variable related to it, and confirm that the result of this variable through infinite process is the unknown quantity sought; Finally, the results are obtained by limit calculation.
The idea of limit is the basic idea of calculus, and a series of important concepts in mathematical analysis such as continuity, derivative and definite integral of function are defined by means of limit. If you want to ask, "What is the theme of mathematical analysis?" Then it can be summed up as follows: "Mathematical analysis is a subject that studies functions with extreme ideas".
The origin of emergence and development (1).
Like all scientific thinking methods, extreme thinking is also the product of social practice. The thought of limit can be traced back to ancient times, and Liu Hui's cyclotomy is an application of the original limit thought based on intuition. The ancient Greeks' exhaustive method also contained the idea of limit, but because of their "fear of infinity", they obviously avoided "taking limit" and completed the relevant proof with the help of indirect proof-reduction to absurdity.
/kloc-in the 6th century, Steven, a Dutch mathematician, improved the ancient Greek exhaustive method in the process of investigating the center of gravity of a triangle. With the help of geometric intuition, he boldly used limit thought to think about problems and gave up the proof of reduction law. In this way, he inadvertently "pointed out the direction of the development of the limit method into a practical concept."
(2) development
The further development of limit thought is closely related to the establishment of calculus. /kloc-in the 0/6th century, Europe was in the embryonic stage of capitalism and its productive forces developed greatly. A large number of problems in production and technology cannot be solved by elementary mathematics alone. Mathematics is required to break through the scope of traditional research constants and provide new tools that can be used to describe and study the process of motion change. This is the social background to promote extreme development and establish calculus.
At first, Newton and Leibniz established calculus based on the concept of infinitesimal. Later, due to logical difficulties, everyone accepted the idea of limit to varying degrees in their later years. Newton expressed the average velocity of a moving object by the ratio of distance change Δ s to time change Δ t, which made Δ t infinitely close to zero, thus obtaining the instantaneous velocity of the object, and thus leading to the concepts of derivative and differential theory. He realized the importance of the concept of limit and tried to use it as the basis of calculus. He said: "If two quantities and the ratio of quantities remain equal for a limited time and are close to each other before the end of this time, so that the difference is less than any given difference, then it will eventually become equal." But Newton's concept of limit is also based on geometric intuition, so he can't get a strict limit expression. Newton's concept of limit is only close to the following intuitive language description: "If N is infinitely increased and an is infinitely close to the constant A, it is said that an takes A as the limit".
This descriptive language is easily accepted by people, and this definition is often used in some modern elementary calculus reading materials. However, this definition does not quantitatively give the relationship between two "infinite processes" and cannot be used as the logical basis of scientific argumentation.
It was precisely because of the lack of strict limit definition at that time that calculus theory was suspected and attacked by people. For example, in the concept of instantaneous velocity, is δ t equal to zero? If it is zero, how to divide it? If it is not zero, how to delete those items that contain it? This is the infinitesimal paradox in the history of mathematics. British philosopher and archbishop Becquerel attacked calculus the most violently. He said that the derivation of calculus was "obvious sophistry".
Becker's fierce attack on calculus, on the one hand, serves religion; On the other hand, due to the lack of a solid theoretical foundation of calculus at that time, even Newton himself could not get rid of the confusion in the concept of limit. This fact shows that understanding the concept of limit and establishing a strict theoretical foundation of calculus are not only necessary for mathematics itself, but also of great significance in epistemology.
(3) perfection
The perfection of limit thought is closely related to the rigor of calculus. For a long time, many people have tried to solve the theoretical problems of calculus, but they have failed. This is because the research object of mathematics has expanded from constant to variable, and people are not very clear about the unique law of variable mathematics; There is still a lack of understanding of the differences and connections between variable mathematics and constant mathematics; The unity of opposites between finite and infinite is still unclear. In this way, people can't adapt to the new needs of variable mathematics with the traditional thinking method of dealing with constant mathematics, nor can they just use the old concepts to explain the dialectical relationship between "zero" and "non-zero".
/kloc-in the 8th century, Robbins, D'Alembert, Lorrell and others clearly put forward that limit must be the basic concept of calculus, and they all made their own definitions of limit. D'Alembert's definition is: "One quantity is the limit of another quantity, if the second quantity is closer to the first quantity than any given value", which is close to the correct definition of limit; However, none of these people's definitions can get rid of their dependence on geometric intuition. This is the only way, because most of the concepts of arithmetic and geometry before19th century are based on the concept of geometric quantities.
Czech mathematician Porzano first gave the correct definition of derivative with the concept of limit. He defined the derivative of the function f(x) as the limit f'(x) of the difference quotient Δ y/Δ x, and he emphasized that f'(x) is not the quotient of two zeros. Porzano's thought is valuable, but he still hasn't figured out the essence of limit.
/kloc-in the 9th century, French mathematician Cauchy expounded the concept and theory of limit completely on the basis of predecessors' work. In the process of analysis, he pointed out: "When the value of a variable infinitely approaches a fixed value, the difference between the value of the variable and the fixed value is as small as possible, and this fixed value is called the limit value of all other values. Especially, when the value (absolute value) of a variable is infinite,
Cauchy regards infinitesimal as a variable with a limit of 0, and clarifies the fuzzy understanding that infinitesimal is "zero-like non-zero", that is, in the process of change, its value can be non-zero, but its changing trend is "zero" and it can be infinitely close to zero.
Cauchy tried to eliminate the geometric intuition in the concept of limit, clarify the definition of limit, and then fulfill Newton's wish. However, there are still descriptive words such as "infinite approximation" and "the smaller the better" in Cauchy's narrative, so there are still intuitive traces of geometry and physics, which are not completely rigorous.
In order to eliminate the intuitive trace in the concept of limit, Wilstrass put forward the static definition of limit, which provided a strict theoretical basis for calculus. The so-called an=A means: "If there is ε >; 0, there is always a natural number n, so when n >; When n, inequality | an-a |
With the help of inequality, this definition quantitatively and concretely describes the relationship between two "infinite processes" through the relationship between ε and n, so this definition is strict and can be used as the basis of scientific argumentation, and it is still used in mathematical analysis books. This definition only involves the number and its size relationship, and only involves the words given, existing, arbitrary, etc., getting rid of the word "close" and no longer resorting to the intuition of movement.
As we all know, constant mathematics is a static study of mathematical objects. Since the emergence of analytic geometry and calculus, movement has entered mathematics, and it is possible for people to study physical processes dynamically. After that, the ε-N language established by Wilstras describes the changing trend of variables with static definitions. This spiral evolution of "static-dynamic-static" reflects the dialectical law of mathematical development.
2. The thinking function of extreme thinking
Extreme thinking has been widely used in modern mathematics and even physics, which is determined by its inherent thinking function. The thought of limit reveals the unity of opposites between variables and constants, infinity and finiteness, which is the application of the law of unity of opposites of materialist dialectics in the field of mathematics. With the help of limit thought, people can know infinity from finiteness, change from invariability, curve from straight line, qualitative change from quantitative change and accuracy from approximation.
Infinite and finite are essentially different, but they are related. Infinity is a limited development. The sum of infinite numbers is not a general algebraic sum. Defining it as the limit of "partial sum" is to understand infinity from finiteness with the help of the thinking method of limit.
"Change" and "invariance" reflect two different states of things, namely, motion change and relative static, but they can be transformed into each other under certain conditions, which is "one of the powerful levers of mathematical science". For example, it is impossible to solve the instantaneous velocity of linear motion with variable speed by elementary method. The difficulty is that the speed is variable. For this reason, people first use uniform speed instead of variable speed in a small range to find its average speed, and define instantaneous speed as the limit of average speed, that is, with the help of limit thinking method, we can understand "change" from "invariability"
There are essential differences between curves and straight lines, but they can also be transformed into each other under certain conditions. As Engels said, "Straight lines and curves are finally equal in differentiation." Making good use of this unity of opposites is one of the important means to deal with mathematical problems. The area of a straight line is easy to get, but the problem of finding the area of a curve cannot be solved by elementary methods. Liu Hui approximates a circle with polygons inscribed in it. Generally speaking, people approach the area of a curved trapezoid with the area of a small rectangle, and they all know the curved shape from a straight line with the help of the limit thinking method.
There are differences and connections between quantitative change and qualitative change, and there is a dialectical relationship between them. Quantitative change can cause qualitative change. The mutual change between quality and quantity is one of the basic laws of dialectics and plays an important role in mathematical research. For any circle inscribed with a regular polygon, when the number of sides is doubled, it will still be inscribed with a regular polygon, which is quantity rather than quality; However, if the number of sides is doubled continuously, the polygon will "change" into a circle after an infinite process, and the polygon area will be converted into a circular area. This is to understand qualitative change from quantitative change with the help of extreme thinking methods.
Approximation and accuracy are unity of opposites, and they can also be transformed into each other under certain conditions, which is an important measure for mathematics to be applied to practical calculation. The aforementioned "partial sum", "average speed" and "area of a circle inscribed in a regular polygon" are approximate values of the corresponding "sum of infinite series", "instantaneous speed" and "area of a circle" respectively, and the corresponding accurate values can be obtained after taking the limit. This is all with the help of extreme thinking method, from approximate understanding to accuracy.