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Fundamental Theorems on Limits

 

Theorem I. The limit of the algebraic sum of finitely many functions equals the sum of the limits of these functions:

 

.

Theorem II. The limit of the product of two functions equals the product of the limits of these functions:

.

Theorem III. The limit of the ratio of two functions equals the quotient of the limits of the numerator and the denominator:

.

 

Computations of limits. Examples.

I. Limits as x.

(1)

 

The limits in the numerator and the denominator equal zero.

To find the limit of a linear-fractional function, we must divide the numerator and the denominator by to the maximum power among the powers of x in the numerator and the denominator.

(2)

because 4 is the maximum power of x in the numerator and the denominator.

(3) (divide by 2).

A simple method for finding limits of linear-fractional functions as  is to leave the term containing the maximum power of in the numerator and the denominator:

4) ,

 

5) ,

 

6) .

 

Let us find limits (1), (2), (3) by the simple method:

,

 

,

 

.

 

Deleting the terms containing lower powers of x from the numerator and the denominator is only possible because, after division by to the maximum power, the limits of all such terms vanish.

II. Limits as . Looking for a limit, first, substitute in the function. If we obtain a number, then this number is the limit of the function. If we obtain one of the indeterminacies ,1, and , then we must eliminate it by transforming the function and then to pass to the limit.

 

(1) ,

 

 

(2) ,

 

(3) ,

 

The First Remarkable Limit and Its Generalization

 

The following limit exists and equals 1:

 

.

 

Example:

(1) ,

2) .

 

The first generalized remarkable limit. The first remarkable limit can be generalized, namely, written in the more general form

(4)

 

In this formula, () is an infinitesimal; it is very important that the argument of the sine and the denominator must be absolutely identical.

Examples.

(1) ,

 

(2) .

 

The Second Remarkable Limit

 

 

Consider the limit

. (5)

 

The number satisfies the inequalities 2<e<3, e is an abbreviation for exponentials, i.e., outer; it is sometimes denoted by e=exp and approximately equals e2,71828. y=ex is the exponential function.

Examples. Find the following limits by using the second remarkable limit:

 

.

 

The Second Generalized Remarkable Limit

 

The second remarkable limit (*) and its modification (**) can be generalized, i.e., written in the more general forms

 

and

. (6)

 

In these formulas, () is an infinitesimal and N(x) is an infinitude. It is very important that in these formulas, N(x) and () are absolutely identical in the denominators and exponents.



For example,

Other Remarkable Limits

 

Consider the following limits of functions often encountered in applications:

, (7)

for a=e, . (8)

. (9)

 

 


Date: 2015-01-02; view: 1619


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Function. function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities. The ends | The derivative of the function. Geometric and mechanical meaning. Table of derivatives. The differential of a function
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