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Function. function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities. The endsLECTURE PLAN:
1. Function limit 2. Infinitely small and infinitely large quantities 3. Main theorems about limits and their applications 4. Continuity of function
The limit of a sequence. Definition. A sequence is an infinite set of terms, each of which is assigned a number. The terms of a sequence must obey a certain law. õ1,õ2,õ3,õ4,…,õn,…. Example.
Definition. A number à is called the limit of a sequence if, for any >0, there exists a number N() depending on , such
Notation:
Example.
Definition. The limit of a variable õ is a number à such that for any >0, there exists an õ starting with which all õ satisfy the inequalities
Notation:
à– à à+ õ(à–;à+) Properties: 1. The limit of a constant number equals this number. 2. A variable can not have two different limits. 3. Some variables have no limit. Example. Definition. We say that õ tends to infinity if, for any number Ì, there exists an õ such that, starting it,
(a) M>0, x>M, x; (b) M<0, x<–M, x –. Example. xn=n2+1 ; as n, õn tends to infinity. The limit of a function. Suppose that y=f(x) is a function defined on a domain D containing a point à: àD. Definition. A number b is called the limit of the function f(x) as õ à if, for any given >0, there exists a small positive depending on (()>0) such that, for any õ satisfying the inequality
Example. Find the limit of f(x)=5x–1 as x2, and determine .
i.e., To find , we must find x from the inequality for the function and substitute it in the inequality for the variable. Definition. The left limit of a function f(x) as x a is the limit of f(x) as x a, and õ<à. Notation:
Definition. The right limit of a function f(x) as x a, is the limit of f(x) as x a, and õ>à. Notation:
If the left limit equals the right limit and some number b, then b is the limit of the function as x a. Example.
Definition. A number b is called the limit of f(x) as õ if, for any >0, there exists a (large) number N depending on such that Notation:
Infinitesimals and bounded functions. Definition. A function f(x) is said to be infinitesimals as õ à if, for any Ì, there exists a such that
Notation:
Definitions. 1. A function f(x) is said to be bounded on a domain D if, for any õ from D, |f(x)|M.
For example, f(x)=cosx , |cosx|1, M=1, If this condition is not satisfied, then the function is said to be unbounded, i.e., this function is an infinite quantity. 2. A function f(x) is said to be bounded as õ à if, for any õ from a neighborhood of à, |f(x)|M. 3. A function f(x) is said to be bounded as õ if, for any õ>N, |f(x)|M. Theorem I. If a function f(x) has a finite limit as xa then f(x) is bounded as x a. Theorem II. If a function has a limit as õ à and this limit is not equal to zero, then
The infinitesimals and their properties. Definition 1. A function (õ) is called an infinitesimal as õ à if
Definition 2. A function (õ) is called an infinitesimal as õ à if, for all õ( à-; à+),
These two definitions are equivalent, i.e., we can obtain the second definition from the first and vice versa. (Prove this). Theorem I. If a function f(x) is represented as the sum of a constant number and an infinitesimal, i.e., f(x)=b+(x), (2) then it has a limit:
Conversely, if a function f(x) has limit b, then the function can be represented in the form (2). Theorem II. If (õ) is an infinitesimal as õ à, then Theorem III. The sum of finitely many infinitesimals is an infinitesimal: 1(õ) + 2(õ) + 3(õ) + … + ê(õ)= (õ). Theorem IV. The product (x).z(x) of an infinitesimal (õ) by a bounded function z(x) as x a is an infinitesimal. Corollary. The product of infinitesimals is an even smaller quantity. Theorem V. An infinitesimal divided by a function having nonzero limit as õ à is infinitesimal, i.e., if
Date: 2015-01-02; view: 2388 |