Function. function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities. The ends
1. Function limit
2. Infinitely small and infinitely large quantities
3. Main theorems about limits and their applications
4. Continuity of function
The Theory of Limits
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.
Definition. A number à is called the limit of a sequence if, for any >0, there exists a number N() depending on , such
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
à– à à+
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. . Assigning integer values to n, we obtain: etc.; i.e. this variable has no limit.
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 , . Notation:
Example. Find the limit of f(x)=5x–1 as x2, and determine .
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.
; an indeterminacy, although the limit exists:
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 for any .
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 whenever .
Definitions. 1. A function f(x) is said to be bounded on a domain D if, for any õ from D,
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 is bounded as õ à.
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.,
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 is an infinite quantity.
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