15.2 限制说明

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  Introduction
  ::导言

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  The total worldwide box-office receipts for a blockbuster movie can be approximated by the function  $\begin{array}{r}f(x)=\frac{120x^2-10}{4x^2+1},\end{array}$  where $\begin{array}{r}f(x)\end{array}$  is measured in millions of dollars, and  $\begin{array}{r}x\end{array}$  is the number of months since the movie's release. D etermine the movie's gross earnings for the long run.  A s  $\begin{array}{r}x\end{array}$ gets extremely large, the function $\begin{array}{r}f(x)\end{array}$   approaches   $\begin{array}{r}\frac{120}{4}=30,\end{array}$  because the greatest powers are equal and $\begin{array}{r}\frac{120}{4}\end{array}$  is the ratio of the leading coefficients. However, how is this statement represented using limit notation?
  ::以 F(x) = 120x2 - 104x2+1 函数可以近似于世界范围的硬盘电影的票房收入总额,F(x) = 120x2 - 104x2+1, 其中f(x) 以百万美元计量,x 是电影发行后的月数。确定电影的长期毛收入。当x 变得非常大时,函数 f(x) 接近1204=30, 因为最大功率是相等的, 1204 是主要系数的比重。 然而, 该语句如何使用限制符号表示 ?

  

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  Limit Notation
  ::限制评分

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  When learning about the end behavior of a rational function, you describe the function as either having a horizontal asymptote at some number, or going to infinity. Limit notation is a way of describing this end behavior mathematically.  It  is a way of expressing the fact that the function gets arbitrarily close to a value.
  ::当学习一个理性函数的结束行为时, 您可以描述该函数, 描述该函数不是在某个数字上具有水平的零点, 就是要进入无穷状态。 限制符号是数学描述此结束行为的一种方式。 这是表达该函数任意接近某个值的一种方式 。

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     Limit Notation
  ::限制评分

  The limit of $\begin{array}{r}f\end{array}$ of $\begin{array}{r}x\end{array}$  as  $\begin{array}{r}x\end{array}$  approaches  $\begin{array}{r}a\end{array}$  is  $\begin{array}{r}b\end{array}$ .

  $$\begin{array}{r}\underset{x\to a}{lim}f(x)=b\end{array}$$

   

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  Notice that t he function $\begin{array}{r}f(x)\end{array}$ is any function in terms of  $\begin{array}{r}x\end{array}$ . T he notion of  $\begin{array}{r}x\end{array}$  approaching $\begin{array}{r}a\end{array}$  means  $\begin{array}{r}x\end{array}$  is getting sufficiently close to  $\begin{array}{r}a,\end{array}$  but $\begin{array}{r}x\ne a\end{array}$ . Sufficiently close consists of the values that are less than  $\begin{array}{r}a\end{array}$  (to the left of  $\begin{array}{r}a\end{array}$ ) and greater  than  $\begin{array}{r}a\end{array}$  (to the right of  $\begin{array}{r}a\end{array}$ ).  The letter  $\begin{array}{r}a\end{array}$  can be any number or infinity.
  ::注意函数 f( x) 是 x 中的任何函数 。 x 接近手段 x 的概念已经足够接近于 a, 但是 x\\ {a。 足够接近的值包括小于 ( a 左侧) 和大于 ( a 右侧) 的值。 字母 a 可以是任意数字或无限值 。

  

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  The letter   $\begin{array}{r}b\end{array}$   can only be a number.  If the function goes to infinity, then the function has no limit because infinity is technically not a number . In this case, you should write that the limit does not exist , or DNE . 
  ::字母b 只能是一个数字。如果函数进入无限,则该函数没有限制,因为无限从技术上讲不是一个数字。在这种情况下,您应该写下该限制不存在,或者 DNE 。

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  While a function may never actually reach the value of   $\begin{array}{r}b\end{array}$ , it will get arbitrarily close to  $\begin{array}{r}b\end{array}$ . One way to think about the concept of a limit is to use a physical example. If you stand some distance away from a wall, then take a big step to get halfway to the wall. Take another step to go halfway to the wall again. If you keep taking steps that take you halfway to the wall, then two things will happen. First, you will get extremely close to the wall but never actually reach the wall regardless of how many steps you take. Second, an observer who wishes to describe your situation would notice that the wall acts as a limit to how far you can go.
  ::虽然函数可能永远不会真正达到 b 的值,但它会任意接近 b。 思考限制概念的一种方式是使用物理实例。 如果您离墙有一段距离, 那么就迈出一大步到墙的另一边。 再跨一步到墙的另一边。 如果您继续采取步骤, 将你带到墙的另一边, 那么就会发生两件事。 首先, 你将会非常接近墙, 但实际上永远也不会到达墙, 无论你走多少步。 其次, 观察者如果想描述你的情况, 就会注意到, 墙的作用是限制你走多远。

   

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  Play, Learn, and Explore Limit Definitions: 
  ::游戏、学习和探索限制定义:

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  Examples
  ::实例

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  Example 1
  ::例1

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  Translate the following statement into limit notation: The limit of  $\begin{array}{r}y=4x^2\end{array}$ as  $\begin{array}{r}x\end{array}$  approaches 2 is 16.
  ::将以下语句翻译为限值符号:y=4x2作为x方针2的极限为16。

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  Solution: 
  ::解决方案 :

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  $\begin{array}{r}\underset{x\to 2}{lim}4x^2=16\end{array}$
  ::limx 24x2=16

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  Example 2
  ::例2

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  Translate the mathematical statement below into words.
  ::将下面的数学语句转换为单词。

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  $$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}\underset{i=1}{\overset{n}{\sum }}{\left(\frac{1}{2}\right)}^i=1\end{array}$$

  
  ::limni=1n( 12) i=1

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  Solution:
  ::解决方案 :

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  The limit of the sum of  $\begin{array}{r}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots \end{array}$  as the number of terms approaches infinity is 1.
  ::12+14+18++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++12++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++12+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++12++++++++++++++++++++++++++++++++++++++++12+12+12++++++++++++++++++++12+12+12+12+12+12+12+12+12+12+12+12++++++12+++++12+12+12+++++++++++++++++++++++++++++++++++++++++++++++++++++++12+12+12++12++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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  Example 3
  ::例3

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  Use limit notation to represent the following mathematical statement:
  ::使用限制符号表示以下数学语句:

  $\begin{array}{r}\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots =\frac{1}{2}.\end{array}$

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  Solution:  
  ::解决方案 :

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  $\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}\underset{i=1}{\overset{n}{\sum }}{\left(\frac{1}{3}\right)}^i=\frac{1}{2}\end{array}$
  ::limni=1n( 13) i=12

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  Example 4
  ::例4

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  Recall the question from the Introduction: The  total worldwide box-office receipts for a blockbuster movie can be approximated by the function  $\begin{array}{r}f(x)=\frac{120x^2-10}{4x^2+1},\end{array}$  where  $\begin{array}{r}f(x)\end{array}$  is measured in millions of dollars, and  $\begin{array}{r}x\end{array}$  is the number of months since the movie's release. As  $\begin{array}{r}x\end{array}$ approaches infinity , the function  $\begin{array}{r}f(x)\end{array}$   approaches   $\begin{array}{r}\frac{120}{4}=30\end{array}$ . However, how is this statement represented using limit notation?  
  ::回顾导言中的问题:一个大块头电影的全球票房收入总额可以用函数f(x)=120x2-104x2+1来比较,其中f(x)以百万美元计量,x是电影发行后的月数。作为形形形色色无穷,函数f(x)接近1204=30。然而,该语句如何使用限值符号表示?

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  Solution:
  ::解决方案 :

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  The limit of $\begin{array}{r}f(x)=\frac{120x^2-10}{4x^2+1}\end{array}$  as  $\begin{array}{r}x\end{array}$   approaches infinity is $\begin{array}{r}\frac{120}{4}=30\end{array}$  can be written using limit notation as
  ::f( x) = 120x2 - 104x2+1 的限值, 因为 x 方法的无限度为 1204= 30 , 使用限制符号可以写入 。

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  $$\begin{array}{r}\underset{x\to \mathrm{\infty }}{lim}\left(\frac{120x^2-10}{4x^2+1}\right)=\frac{120}{4}=30.\end{array}$$

  
  ::limx( 120x2 - 104x2+1) = 1204=30。

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  Example 5
  ::例5

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  Describe the end behavior of the following rational function at infinity and negative infinity using limits: 
  ::描述以下使用限制的无限和负无限合理函数的结束行为:

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  $\begin{array}{r}f(x)=\frac{-5x^3+4x^2-10}{10x^3+3x^2+98}.\end{array}$
  ::f(x) 5x3+4x2-1010x3+3x2+98。

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  Solution :
  ::解决方案 :

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  Since the function has equal powers of  $\begin{array}{r}x\end{array}$  in the numerator and in the denominator, the end behavior is $\begin{array}{r}-\frac{1}{2}\end{array}$  as $\begin{array}{r}x\end{array}$  goes to both positive and negative infinity.
  ::由于该函数在分子和分母中具有x的同等权力,所以最终行为为-12,因为x既等于正,又等于负无穷。

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  $\begin{array}{r}\underset{x\to \mathrm{\infty }}{lim}\left(\frac{-5x^3+4x^2-10}{10x^3+3x^2+98}\right)=\underset{x\to -\mathrm{\infty }}{lim}\left(\frac{-5x^3+4x^2-10}{10x^3+3x^2+98}\right)=-\frac{1}{2}\end{array}$
  :- 5x3+4x2- 1010x3+3x2+98) = limx(- 5x3+4x2- 1010x3+3x2+98) = limx(- 5x3+4x2- 1010x3+3x2+98) 12

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  Example 6
  ::例6

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  Translate the following limit expression into words:
  ::将以下限制表达式翻译为单词:

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  $\begin{array}{r}\underset{h\to 0}{lim}\left(\frac{f(x+h)-f(x)}{h}\right)=x.\end{array}$
  ::limh0(f(x+h)-f(x)h)=x。

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  Solution:
  ::解决方案 :

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  The limit of the ratio of the difference between  $\begin{array}{r}f\end{array}$  of quantity $\begin{array}{r}x\end{array}$ plus $\begin{array}{r}h\end{array}$ and $\begin{array}{r}f\end{array}$ of $\begin{array}{r}x\end{array}$ and  $\begin{array}{r}h\end{array}$  as $\begin{array}{r}h\end{array}$ approaches 0 is $\begin{array}{r}x\end{array}$ .
  ::数量x+ h和f之间的差数之差的极限值为xxxx+ h和fx和h之间的差数,因为h接近0是x。

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  Example 7
  ::例7

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  What do you notice about the limit expression in Example 6 ?
  ::关于例6中的限制表达式,您注意到什么?

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  Solution :
  ::解决方案 :

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  You should notice that $\begin{array}{r}h\to 0\end{array}$ does not mean $\begin{array}{r}h=0,\end{array}$ because if it did you could not have a 0 in the denominator. You should also note that in the numerator, $\begin{array}{r}f(x+h)\end{array}$ and $\begin{array}{r}f(x)\end{array}$ are going to be super close together as $\begin{array}{r}h\end{array}$ approaches 0. Calculus will enable you to deal with problems that seem to look like $\begin{array}{r}\frac{0}{0}\end{array}$ and $\begin{array}{r}\frac{\mathrm{\infty }}{\mathrm{\infty }}\end{array}$ .
  ::您应该注意, h0 并不意味着 h=0, 因为如果它不能在分母中为 0。 您也应该注意, 在分子中, f( x+h) 和 f( x) 将会在 h 接近 0 时超级接近 。 计算将使您能够处理看上去像是 00 和 \ 的问题 。

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  Summary
  ::摘要

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  • Limit notation is a way of expressing the fact that the function gets arbitrarily close to a value.
    ::限制评分是一种表达以下事实的方式:函数任意接近某一价值。
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  • The limit of $\begin{array}{r}f\end{array}$ of $\begin{array}{r}x\end{array}$  as  $\begin{array}{r}x\end{array}$  approaches  $\begin{array}{r}a\end{array}$  is  $\begin{array}{r}b\end{array}$  is written as  $\begin{array}{r}\underset{x\to a}{lim}f(x)=b\end{array}$ .
    ::x 的 f 限制值为 x 接近 a b 的 x 限制值,以 limx+*af(x) =b 写成。
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    If the function goes to infinity, then the limit does not exist, or DNE. 
    ::如果该函数进入无限,则该限制不存在,或 DNE。

     

    
    ::如果该函数进入无限,则该限制不存在,或 DNE。

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  Review
  ::回顾

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  Describe the end behavior of the following rational functions at infinity and negative infinity using limits:
  ::描述以下使用限制的无限和负无限合理函数的结束行为:

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  1.  $\begin{array}{r}f(x)=\frac{2x^4+4x^2-1}{5x^4+3x+9}\end{array}$
  ::1. f(x) = 2x4+4x2 - 15x4+3x+9

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  2. $\begin{array}{r}g(x)=\frac{8x^3+4x^2-1}{2x^3+4x+7}\end{array}$
  ::2. g(x)=8x3+4x2-12x3+4x+7

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  3. $\begin{array}{r}f(x)=\frac{x^2+2x^3-3}{5x^3+x+4}\end{array}$
  ::3. f(x)=x2+2x3-3-53x3+x+4

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  4. $\begin{array}{r}f(x)=\frac{4x+4x^2-5}{2x^2+3x+3}\end{array}$
  ::4. f(x)=4x+4x2-52x2+3x+3

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  5. $\begin{array}{r}f(x)=\frac{3x^2+4x^3+4}{6x^3+3x^2+6}\end{array}$
  ::5. f(x)=3x2+4x3+46x3+3x2+6

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  Translate the following statements into limit notation:
  ::将以下语句翻译为限制编号:

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  6. The limit of $\begin{array}{r}y=2x^2+1\end{array}$  as  $\begin{array}{r}x\end{array}$  approaches 3 is 19.
  ::6. y=2x2+1作为x方针3的极限为19。

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  7. The limit of $\begin{array}{r}y=e^x\end{array}$  as  $\begin{array}{r}x\end{array}$  approaches negative infinity is 0.
  ::7. y=ex作为x接近负无限值的极限为0。

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  8. The limit of $\begin{array}{r}y=\frac{1}{x}\end{array}$  as  $\begin{array}{r}x\end{array}$  approaches infinity is 0.
  ::8. y=1x作为x方针的无限限为0。

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  Use limit notation to represent the following mathematical statements:
  ::使用限制符号表示下列数学语句:

  9.  $\begin{array}{r}\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots =\frac{1}{3}\end{array}$

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  10. The series $\begin{array}{r}1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots \end{array}$ diverges.
  ::10. 1+12+13+14系列有差异。

  11. $\begin{array}{r}1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =2\end{array}$

  12. $\begin{array}{r}\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\cdots =1\end{array}$

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  Translate the following mathematical statements into words:
  ::将以下数学语句翻译为单词:

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  13. $\begin{array}{r}\underset{x\to 0}{lim}\frac{5x^2-4}{x+1}=-4\end{array}$
  ::13. 立方 5x2 - 4x+1+14

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  14. $\begin{array}{r}\underset{x\to 1}{lim}\frac{x^3-1}{x-1}=3\end{array}$
  ::14. 立方厘米1x3-一x-1=3

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  15. If $\begin{array}{r}\underset{x\to a}{lim}f(x)=b\end{array}$ , is it possible that $\begin{array}{r}f(a)=b\end{array}$ ?  Explain.
  ::15. 如果(x)=b,f(a)=b?解释一下。

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  Review (Answers )
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。