节: 限制评分 | 14.1 限制说明

章节大纲

When learning about the end behavior of a rational function you described the function as either having a horizontal asymptote at zero or another number, or going to infinity. Limit notation is a way of describing this end behavior mathematically.
::当学习理性函数的结束行为时, 您描述该函数时, 您将函数描述为要么在零位或其它数字上水平小数, 要么是无穷无尽。限制符号是数学描述此结束行为的一种方法。

You already know that as $\begin{aligned} x \end{aligned}$ gets extremely large then the function $\begin{aligned} f (x) = \frac{8 x^{4} + 4 x^{3} + 3 x^{2} - 10}{3 x^{4} + 6 x^{2} + 9 x} \end{aligned}$ goes to $\begin{aligned} \frac{8}{3} \end{aligned}$ because the greatest powers are equal and $\begin{aligned} \frac{8}{3} \end{aligned}$ is the ratio of the leading coefficients. How is this statement represented using limit notation?
::您已经知道, 当 x 变得非常大时, 函数 f( x) = 8x4+4x3+3x2 - 103x4+6x2+9x 就会变成83, 因为最大功率相等, 83 是主要系数的比。这个语句如何使用限制符号表示 ?

Introduction to Limits
::限制的导言

Limit notation is a way of stating an idea that is a little more subtle than simply saying $\begin{aligned} x = 5 \end{aligned}$ or $\begin{aligned} y = 3 \end{aligned}$ .
::限定符号是一种表达一种比简单地说 x=5 或 y=3 更微妙的想法的方式。

$\begin{aligned} lim_{x \to a} f (x) = b \end{aligned}$
::limxaf(x)=b

“The limit of $\begin{aligned} f \end{aligned}$ of $\begin{aligned} x \end{aligned}$ as $\begin{aligned} x \end{aligned}$ approaches $\begin{aligned} a \end{aligned}$ is $\begin{aligned} b \end{aligned}$ ”
::“xx作为x方针a为b的f的限度为b”

The letter $\begin{aligned} a \end{aligned}$ can be any number or infinity. The function $\begin{aligned} f (x) \end{aligned}$ is any function of $\begin{aligned} x \end{aligned}$ . The letter $\begin{aligned} b \end{aligned}$ can be any number. If the function goes to infinity, then instead of writing “ $\begin{aligned} = \infty \end{aligned}$ ” you should write that the limit does not exist or “ DNE ”. This is because infinity is not a number. If a function goes to infinity then it has no limit.
::字母 a 可以是任意数字或无穷无尽。函数 f( x) 是 x 的任何函数。字母 b 可以是任意数字。如果函数转到无穷无穷, 那么您应该写“ ” , 而不是写“ ” , 而不是写“ DNE ” 。这是因为无穷无尽不是数字。如果函数转到无穷无尽, 那么它就没有限制。

Take the following limit:
::以下列限制为限:

The limit of $\begin{aligned} y = 4 x^{2} \end{aligned}$ as $\begin{aligned} x \end{aligned}$ approaches 2 is 16
::y=4x2 的极限值, 如xx 方针2为 16

In limit notation, this would be:
::在限值符号中,这将是:

$\begin{aligned} lim_{x \to 2} 4 x^{2} = 16 \end{aligned}$
::limx 24x2=16

While a function may never actually reach a height of $\begin{aligned} b \end{aligned}$ it will get arbitrarily close to $\begin{aligned} b \end{aligned}$ . One way to think about the concept of a limit is to use a physical example. Stand some distance from a wall and 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.
::虽然函数实际上可能永远不会达到b的高度,但它会任意地接近b。思考限制概念的一种方式是使用物理实例。与墙保持一定距离, 然后迈出一大步才能到墙的另一边。再跨一步再跨到墙的另一边。如果您继续采取将你带到墙的另一边的步骤, 就会发生两件事。首先, 你将非常接近墙, 但实际上永远不会到达墙, 无论你走多少步。其次, 观察员如果想描述你的情况, 就会注意到, 墙是限制你走多远的限度。

Examples
::实例

Example 1
::例1

Earlier, you were asked how to write the statement "The limit of $\begin{aligned} \frac{8 x^{4} + 4 x^{3} + 3 x^{2} - 10}{3 x^{4} + 6 x^{2} + 9 x} \end{aligned}$ as $\begin{aligned} x \end{aligned}$ approaches infinity is $\begin{aligned} \frac{8}{3} \end{aligned}$ " in limit notation.
::早些时候,有人问您如何在限制符号中写“8x4+4x3+3x2-103x4+6x2+9xx作为x无穷度为83”的语句。

This can be written using limit notation as:
::可以用下列限制符号写成 :

$\begin{aligned} lim_{x \to \infty} (\frac{8 x^{4} + 4 x^{3} + 3 x^{2} - 10}{3 x^{4} + 6 x^{2} + 9 x}) = \frac{8}{3} \end{aligned}$
::limx( 8x4+4x3+3x2 - 103x4+6x2+9x)=83

Example 2
::例2

Translate the following mathematical statement into words.
::将以下数学语句转换为单词。

$\begin{aligned} lim_{n \to \infty} \sum_{i = 1}^{n} {(\frac{1}{2})}^{i} = 1 \end{aligned}$
::limni=1n( 12) i=1

The limit of the sum of $\begin{aligned} \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \end{aligned}$ 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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Example 3
::例3

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

$\begin{aligned} \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots = \frac{1}{2} \end{aligned}$

$\begin{aligned} lim_{n \to \infty} \sum_{i = 1}^{n} {(\frac{1}{3})}^{i} = \frac{1}{2} \end{aligned}$
::limni=1n( 13) i=12

Example 4
::例4

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的理性函数的结束行为。

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

Since the function has equal powers of $\begin{aligned} x \end{aligned}$ in the numerator and in the denominator, the end behavior is $\begin{aligned} - \frac{1}{2} \end{aligned}$ as $\begin{aligned} x \end{aligned}$ 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。

$\begin{aligned} lim_{x \to \infty} (\frac{- 5 x^{3} + 4 x^{2} - 10}{10 x^{3} + 3 x^{2} + 98}) = lim_{x \to - \infty} (\frac{- 5 x^{3} + 4 x^{2} - 10}{10 x^{3} + 3 x^{2} + 98}) = - \frac{1}{2} \end{aligned}$
:- 5x3+4x2- 1010x3+3x2+98) = limx(- 5x3+4x2- 1010x3+3x2+98) = limx(- 5x3+4x2- 1010x3+3x2+98) 12

Example 5
::例5

Translate the following limit expression into words. What do you notice about the limit expression?
::将以下限制表达式转换为单词。您注意到关于限制表达式的什么了吗 ?

$\begin{aligned} lim_{h \to 0} (\frac{f (x + h) - f (x)}{h}) = x \end{aligned}$
::limh0( f( x+h) - f( x) h=x

The limit of the ratio of the difference between $\begin{aligned} f \end{aligned}$ of quantity $\begin{aligned} x \end{aligned}$ plus $\begin{aligned} h \end{aligned}$ and $\begin{aligned} f \end{aligned}$ of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} h \end{aligned}$ as $\begin{aligned} h \end{aligned}$ approaches 0 is $\begin{aligned} x \end{aligned}$ .
::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。

You should notice that $\begin{aligned} h \to 0 \end{aligned}$ does not mean $\begin{aligned} h = 0 \end{aligned}$ because if it did then you could not have a 0 in the denominator. You should also note that in the numerator, $\begin{aligned} f (x + h) \end{aligned}$ and $\begin{aligned} f (x) \end{aligned}$ are going to be super close together as $\begin{aligned} h \end{aligned}$ approaches zero. Calculus will enable you to deal with problems that seem to look like $\begin{aligned} \frac{0}{0} \end{aligned}$ and $\begin{aligned} \frac{\infty}{\infty} \end{aligned}$ .
::您应该注意到, h0 并不意味着 h=0, 因为如果是h=0, 那么分母中就不会有 0。您也应该注意到, 在分子中, f( x+h) 和 f( x) 将高度接近于 h 接近零。计算将使您能够处理看上去像是 00 和 {} 的问题。

Summary

Limit notation is a way of describing the behavior of a function at a given location $\begin{aligned} a . \end{aligned}$

$\begin{aligned} lim_{x \to a} f (x) = b \end{aligned}$

::限值符号是一种描述函数在特定位置a.limxaf(x)=b的行为方式

In words, this says “The limit of $\begin{aligned} f \end{aligned}$ of $\begin{aligned} x \end{aligned}$ as $\begin{aligned} x \end{aligned}$ approaches $\begin{aligned} a \end{aligned}$ is $\begin{aligned} b \end{aligned}$ ”.
::换句话说,这表示“xx作为x接近a的限度为b”。

If the function goes to infinity, then the limit does not exist (DNE).
::如果该函数进入无限,则该限制不存在(DNE)。

Review
::回顾

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的理性函数的结束行为。

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

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

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

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

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

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

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

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

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

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

9. $\begin{aligned} \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots = \frac{1}{3} \end{aligned}$

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

11. $\begin{aligned} 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 2 \end{aligned}$

12. $\begin{aligned} \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots = 1 \end{aligned}$

Translate the following mathematical statements into words.
::将以下数学语句转换为单词。

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

14. $\begin{aligned} lim_{x \to 1} \frac{x^{3} - 1}{x - 1} = 3 \end{aligned}$
::14. 立方厘米1x3-一x-1=3

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

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

Introduction to Limits ::限制的导言

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)