15.4 确定限额的表格

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  要查找限制的表格

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  Introduction
  ::导言

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  Recall that a limit is the value the function approaches as  $\begin{array}{r}x\end{array}$  gets arbitrarily close to a certain number or infinity. Suppose you are 5 feet away from a fan. Then you take a step halfway closer to the fan. You repeat this process until you are nearly on top of  the fan. However, you will never actually be standing in the same place as the fan. How can this situation be described using a limit and table? 
  ::提醒注意一个限制值是当 x 任意接近某个数字或无限时, 函数接近的值。 假设您离风扇有5英尺远。 然后您向风扇走近半步。 您重复此进程, 直到您接近风扇顶部。 然而, 您永远不会真正站在与风扇同处的位置上。 如何用限制和表格描述这种情况 ?

  

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  Limits from Tables
  ::表格的限额

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  A second way to solve limit problems is to create a table. The  $\begin{array}{r}x\end{array}$ -values in this table are numbers to the left and to the right (on the number line) of the number $\begin{array}{r}x\end{array}$  in the limit. For instance, find the limit of the following:
  ::解决限制问题的第二个办法是创建一个表格。本表格中的 x 值是限制数字 x 的左右数字(在数字行上)。例如,可以找到以下的限值:

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  $$\begin{array}{r}\underset{x\to 4}{lim}\frac{x^2-2x-8}{x-4}.\end{array}$$

  
  ::4x2-2-2x-8x-4。

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  A few $\begin{array}{r}x\end{array}$ -values are close to 4 on the left and on the right include 3.9, 3.99, 3.999, 4.001, 4.01, and 4.1. Plug these   $\begin{array}{r}x\end{array}$ - values into the function to obtain the function value at this point.
  ::左侧和右侧的一些X值接近4,包括3.9、3.99、3.99、3.999、4.001、4.01和4.1。将这些x值插入函数中,以获得此点的函数值。

  $\begin{array}{r}x\end{array}$	3.9	3.99	3.999	4	4.001	4.01	4.1
  $\begin{array}{r}f(x)\end{array}$	5.9	5.99	5.999	?	6.001	6.01	6.1

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  Notice that the function values in the bottom row approach the number 6 from the left and the right. Using this table, you could conclude that the limit of the given function is 6 as $\begin{array}{r}x\end{array}$  approaches 4.
  ::注意底行的函数值接近左右的6号值。使用此表格,您可以得出结论,给定函数的极限是 6 个, 如 x 位数 4 。

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  Another example of this solving approach can be seen in the following video: 
  ::以下视频中可以看到这种解决方法的另一个实例:

    

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  Play, Learn, and Explore Tables to Find  Limits: 
  ::播放、学习和探索表格以查找限制 :

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  Examples
  ::实例

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  Example 1
  ::例1

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  Complete the table and use the result to estimate the limit.
  ::填写表格并使用结果估算限额。

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  $$\begin{array}{r}\underset{x\to 2}{lim}\frac{x-2}{x^2-x-2}\end{array}$$

  
  ::立方厘米x2x-2x2-x2-x-2

  $\begin{array}{r}x\end{array}$	1.9	1.99	1.999	2.001	2.01	2.1
  $\begin{array}{r}f(x)\end{array}$

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  Solution:
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	1.9	1.99	1.999	2.001	2.01	2.1
  $\begin{array}{r}f(x)\end{array}$	0.34483	0.33445	0.33344	0.33322	0.33223	0.32258

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  By this table, the limit is $\begin{array}{r}\frac{1}{3}\end{array}$ .
  ::按本表计算,限额为13。

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  Example 2
  ::例2

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  Complete the table and use the result to estimate the limit.
  ::填写表格并使用结果估算限额。

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  $$\begin{array}{r}\underset{x\to 2}{lim}\frac{x-2}{x^2-4}\end{array}$$

  
  ::limx% 2x-2x2-2- 4

  $\begin{array}{r}x\end{array}$	1.9	1.99	1.999	2.001	2.01	2.1
  $\begin{array}{r}f(x)\end{array}$

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  Solution:
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	1.9	1.99	1.999	2.001	2.01	2.1
  $\begin{array}{r}f(x)\end{array}$	0.25641	0.25063	0.25006	0.24994	0.24938	0.2439

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  By this table, the limit is  $\begin{array}{r}\frac{1}{4}\end{array}$ .
  ::按本表计算,限额为14。

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  Example 3
  ::例3

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  Complete the table and use the result to estimate the limit.
  ::填写表格并使用结果估算限额。

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  $$\begin{array}{r}\underset{x\to 0}{lim}\frac{\sqrt{x+3}-\sqrt{3}}{x}\end{array}$$

  
  ::limx=0x+3-3-3x

  $\begin{array}{r}x\end{array}$	-0.1	-0.01	-0.001	0.001	0.01	0.1
  $\begin{array}{r}f(x)\end{array}$

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  Solution:
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	-0.1	-0.01	-0.001	0.001	0.01	0.1
  $\begin{array}{r}f(x)\end{array}$	0.29112	0.28892	0.2887	0.28865	0.28843	0.28631

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  By this table, the limit is between 0.2887 and 0.28865. When you learn to find the limit analytically , you will know that the exact limit is  $\begin{array}{r}\frac{1}{2}\cdot 3\frac{1}{2}\approx 0.2886751346\end{array}$ .
  ::在本表中,上限介于0.2887至0.288665之间。当您通过分析找到极限时,您就会知道准确的界限是123120.2886751346。

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  Example 4
  ::例4

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  Recall the question from the Introduction: Y ou are 5 feet away from a fan and are taking steps closer to the fan. How can this situation be described using a limit and table? 
  ::回顾导言中的问题:你离风扇只有5英尺远,正在接近风扇采取步骤,如何用限制和表格来描述这种情况?

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  Solution:
  ::解决方案 :

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  T he limit representing this situation would be   $\begin{array}{r}\underset{x\to 0}{lim}f(x)\end{array}$  if you define the function so the fan is at 0 and you are at 5 or -5 to start. In the table, the  $\begin{array}{r}x\end{array}$ -values would be to the left and to the right of 0. 
  
  ::如果您定义此函数, 扇扇为 0, 且您在 5 或 - 5 时要开始, 表示此情形的限值为 limx% 0f( x) 。 在表格中, x 值将位于左对面, 则位于右对面 0 。

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  Example 5
  ::例5

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  Graph the following function and the use a table to verify the limit as $\begin{array}{r}x\end{array}$ approaches 1:
  ::以下图解函数和使用表格来验证x方针1的限值:

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  $$\begin{array}{r}f(x)=\frac{x^3-1}{x-1},x\ne 1.\end{array}$$

  
  :xx) =x3 - 1x - 1,x1。

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  Solution:
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}f(x)\end{array}$
  .75	2.3125
  .9	2.71
  .99	2.9701
  .999	2.997
  1	Error
  1.001	3.003
  1.01	3.0301
  1.1	3.31
  1.25	3.8125

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  By this table, the limit is 3.
  ::按本表计算,限额为3。

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  Example 6
  ::例6

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  Estimate the limit numerically .
  ::以数值估计限额。

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  $$\begin{array}{r}\underset{x\to 2}{lim}\frac{x^2-3x+2}{x-2}\end{array}$$

  
  ::limx%2x2 - 3x+2x-2

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  Solution :
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	1.75	1.9	1.99	1.999	2	2.001	2.01	2.1	2.25
  $\begin{array}{r}f(x)\end{array}$	0.75	0.9	0.99	0.999	Error	1.001	1.01	1.1	1.25

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  By this table, the limit is 1.
  ::按本表计算,限额为1。

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  Example 7
  ::例7

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  Estimate the limit numerically.
  ::以数值估计限额。

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  $$\begin{array}{r}\underset{x\to 0}{lim}\frac{\left[\frac{4}{x+2}\right]-2}{x}\end{array}$$

  
  ::limx%0[4x+2]-2x

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  Solution:
  ::解决方案 :

  $\begin{array}{r}x\end{array}$	-0.1	-0.01	-0.001	0.001	0.01	0.1
  $\begin{array}{r}f(x)\end{array}$	0.20526	0.02005	0.002	-0.002	-0.02	-0.1952

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  By this table, the limit is 0.
  
  
  ::按本表格计算,上限为0。

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  Summary
  ::摘要

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  • To solve limits numerically, use a table with   $\begin{array}{r}x\end{array}$ -values to the left and to the right on the number line to the number  $\begin{array}{r}x\end{array}$  in the limit.   
    ::要用数字解析限制,请使用向左和向右的带有 x 值的表格,数字行的右边与限制的 x 数。
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  • Then plug these   $\begin{array}{r}x\end{array}$ - values into the function to obtain the function value at this point.  
    ::然后将这些 X 值插入函数,以便在此点获得函数值。

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  Review
  ::回顾

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  Estimate the following limits numerically using a table:
  ::使用表格以数字方式估计下列限值:

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  1. $\begin{array}{r}\underset{x\to 5}{lim}\frac{x^2-25}{x-5}\end{array}$
  ::1. limx=5x2-2-25x-5

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  2. $\begin{array}{r}\underset{x\to -1}{lim}\frac{x^2-3x-4}{x+1}\end{array}$
  ::2. limx%1x2-3x-4x+1

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  3. $\begin{array}{r}\underset{x\to 2}{lim}\frac{x^3-5x^2+2x-4}{x^2-3x+2}\end{array}$
  ::3. limx%2x3 - 5x2+2x-4x2 - 3x+2

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  4. $\begin{array}{r}\underset{x\to 0}{lim}\frac{\sqrt{x+2}-\sqrt{2}}{x}\end{array}$
  ::4. limx=0x+2-2-2x

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  5. $\begin{array}{r}\underset{x\to 3}{lim}\left(\frac{1}{x-3}-\frac{9}{x^2-9}\right)\end{array}$
  ::5. 立正3(1x-3-3-9x2-9)

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  6. $\begin{array}{r}\underset{x\to 2}{lim}\frac{x^2+5x-14}{x-2}\end{array}$
  ::6. limx%2x2+5x-14x-2

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  7. $\begin{array}{r}\underset{x\to 1}{lim}\frac{x^2-8x+7}{x-1}\end{array}$
  ::7. limx=1x2-8x+7x-1

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  8. $\begin{array}{r}\underset{x\to 0}{lim}\frac{\sqrt{x+5}-\sqrt{5}}{x}\end{array}$
  ::8. limx=0x+5-5-5x

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  9. $\begin{array}{r}\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}\end{array}$
  ::9. limx%9x-3x-9

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  10. $\begin{array}{r}\underset{x\to 0}{lim}\frac{x^2+5x}{x}\end{array}$
  ::10. limx=0x2+5xxx

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  11. $\begin{array}{r}\underset{x\to -3}{lim}\frac{x^2-9}{x+3}\end{array}$
  ::11. limx=%3x2-9x+3

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  12. $\begin{array}{r}\underset{x\to 4}{lim}\frac{\sqrt{x}-2}{x-4}\end{array}$
  ::12. 立方 4x-2x-4

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  13. $\begin{array}{r}\underset{x\to 2}{lim}\frac{\sqrt{x+3}-2}{x-1}\end{array}$
  ::13. limx%2x+3-2x-1

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  14. $\begin{array}{r}\underset{x\to 5}{lim}\frac{x^2-25}{x^3-125}\end{array}$
  ::14. 立方×5x2-2-25x3-125

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  15. $\begin{array}{r}\underset{x\to -1}{lim}\frac{x-2}{x+1}\end{array}$
  ::15. limx%1x-2x+1

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  Review (Answers )
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。