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Function. function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities. The ends

LECTURE PLAN:

 

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.

1,2,3,4,,n,.

Example.

 

Definition. A number is called the limit of a sequence if, for any >0, there exists a number N() depending on , such

for n>N,

 

Notation:

.

Example.

.

 

Definition. The limit of a variable is a number such that for any >0, there exists an starting with which all satisfy the inequalities

.

Notation:

2

 

 +

(;+)

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. . Assigning integer values to n, we obtain: etc.; i.e. this variable has no limit.

Definition. We say that tends to infinity if, for any number , there exists an such that, starting it,

.

(a) M>0, x>M, x;

(b) M<0, x<M, x .

Example. xn=n2+1 ; as n, n tends to infinity.

The limit of a function. Suppose that y=f(x) is a function defined on a domain D containing a point : D.

Definition. A number b is called the limit of the function f(x) as if, for any given >0, there exists a small positive depending on (()>0) such that, for any satisfying the inequality , . Notation:

. (1)

Example. Find the limit of f(x)=5x1 as x2, 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 <. Notation:

 

.

 

Definition. The right limit of a function f(x) as x a, is the limit of f(x) as x a, and >. 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.

; an indeterminacy, although the limit exists:

.

Definition. A number b is called the limit of f(x) as  if, for any >0, there exists a (large) number N depending on such that for any .

Notation:

.

 

Infinitesimals and bounded functions.

Definition. A function f(x) is said to be infinitesimals as if, for any , there exists a such that whenever .

 

Notation:

.

 

Definitions. 1. A function f(x) is said to be bounded on a domain D if, for any 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 if, for any from a neighborhood of , |f(x)|M.

3. A function f(x) is said to be bounded as   if, for any >N, |f(x)|M.

Theorem I. If a function f(x) has a finite limit as xa then f(x) is bounded as x a.

Theorem II. If a function has a limit as and this limit is not equal to zero, then is bounded as .

 

The infinitesimals and their properties.

Definition 1. A function () is called an infinitesimal as if

.

Definition 2. A function () is called an infinitesimal as if, for all ( -; +),

.

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 () is an infinitesimal as , then is an infinite quantity.

Theorem III. The sum of finitely many infinitesimals is an infinitesimal:

1() + 2() + 3() + + ()= ().

Theorem IV. The product (x).z(x) of an infinitesimal () 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 is infinitesimal, i.e., if

, then is an infinitesimal.

 


Date: 2015-01-02; view: 947


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