АвтоАвтоматизацияАрхитектураАстрономияАудитБиологияБухгалтерияВоенное делоГенетикаГеографияГеологияГосударствоДомДругоеЖурналистика и СМИИзобретательствоИностранные языкиИнформатикаИскусствоИсторияКомпьютерыКулинарияКультураЛексикологияЛитератураЛогикаМаркетингМатематикаМашиностроениеМедицинаМенеджментМеталлы и СваркаМеханикаМузыкаНаселениеОбразованиеОхрана безопасности жизниОхрана ТрудаПедагогикаПолитикаПравоПриборостроениеПрограммированиеПроизводствоПромышленностьПсихологияРадиоРегилияСвязьСоциологияСпортСтандартизацияСтроительствоТехнологииТорговляТуризмФизикаФизиологияФилософияФинансыХимияХозяйствоЦеннообразованиеЧерчениеЭкологияЭконометрикаЭкономикаЭлектроникаЮриспунденкция

Limits- indeterminate forms and L’Hospital’s Rule

Читайте также:
  1. Basic Forms of Business Organisation
  2. Dialectics and Its Historical Forms
  3. Forms of Business Organization
  4. Irregular Forms
  5. IV. Complete the sentences with the comparative forms of adjectives.
  6. IV. Translate into Russian paying attention to the «-ed»-forms.
  7. Syntactic relations in Engl, forms and means of syntactic connection. The word group theory.
  8. Using Short Passive Forms to Describe Procedure
  9. V. Complete the sentences with the comparative forms of adjectives.
  10. Объекты Graphics и Drawing в Windows Forms

I. Indeterminate Form of the Type

We have previously studied limits with the indeterminate form as shown in the following example:

Example:

However, there is a general, systematic method for determining limits with the indeterminate form . Suppose that f and g are differentiable functions at x = a a nd that is an indeterminate form of the type ; that is, and . Since f and g are differentiable functions at x = a, then f and g are continuous at x = a; that is, = 0 and = 0. Furthermore, since f and g are differentiable functions at x = a, then and . Thus, if , then

if and

are continuous at x = a. This illustrates a special case of the technique known as L’Hospital’s Rule.

 

 
 


L’Hospital’s Rule for Form

Suppose that f and g are differentiable functions on an open interval

containing x = a, except possibly at x = a, and that and

. If has a finite limit, or if this limit is or

, then . Moreover, this statement is also true

in the case of a limit as or as

 

 
 


In the following examples, we will use the following three-step process:

 

Step 1. Check that the limit of is an indeterminate form of type . If it is not, then L’Hospital’s Rule cannot be used.

Step 2. Differentiate f and g separately. [ Note: Do not differentiate using the quotient rule! ]

Step 3. Find the limit of . If this limit is finite, , or , then it is equal to the limit of . If the limit is an indeterminate form of type , then simplify algebraically and apply L’Hospital’s Rule again.

 

II. Indeterminate Form of the Type

We have previously studied limits with the indeterminate form as shown in the following example:

Example:

 

 

However, we could use another version of L’Hospital’s Rule.

 

L’Hospital’s Rule for Form

Suppose that f and g are differentiable functions on an open interval

containing x = a, except possibly at x = a, and that and

. If has a finite limit, or if this limit is or

, then . Moreover, this statement is also true

in the case of a limit as or as

 

 
 


III. Indeterminate Form of the Type

 

Indeterminate forms of the type can sometimes be evaluated by rewriting the product as a quotient, and then applying L’Hospital’s Rule for the indeterminate

forms of type or .

Example:

IV. Indeterminate Form of the Type

 

A limit problem that leads to one of the expressions

 

, , ,

 

is called an indeterminate form of type . Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction. However, limits problems that lead to one the expressions

, , ,

are not indeterminate, since the two terms work together (the first two produce a limit of and the last two produce a limit of ). Indeterminate forms of the type can sometimes be evaluated by combining the terms and manipulating the result to produce an indeterminate form of type or .

Example:

 

V. Indeterminate Forms of the Types

Limits of the form frequently give rise to indeterminate forms of the types . These indeterminate forms can sometimes be evaluated as follows:

(1)

(2)

(3)

 

The limit on the right hand side of the equation will usually be an indeterminate limit of the type . Evaluate this limit using the technique previously described. Assume that = L.

(4) Finally, .

Example: Find .

 

This is an indeterminate form of the type . Let . 0.

Thus, .

 


1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

Поиск по сайту:



Все материалы представленные на сайте исключительно с целью ознакомления читателями и не преследуют коммерческих целей или нарушение авторских прав. Студалл.Орг (0.008 сек.)