Course: 1.8 限制和零散 | The Way To Learn

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限制和零用量
Suppose you stand exactly 4 feet from a wall, and begin moving toward the wall by halving the distance remaining with each step. How many steps would it take to actually get to the wall? How far would you walk in the process?
::假设你站在离墙4英尺的地方,开始向墙移动,将每一步所剩距离减半。要实际到达墙要走几步才能真正到达墙上?你走多远?

Limits and Asymptotes
::限制和零用量

Consider the function $\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$ . A graph of this function is shown below.
::考虑函数f(x)=1x。此函数的图示如下。

Notice that as the values of x get larger and larger, the graph gets closer and closer to the x -axis. In terms of the function values, we can say that as x gets larger and larger, f ( x ) gets closer and closer to 0. Formally, this kind of behavior of a function is called a limit. We say that as x approaches infinity, the limit of the function is 0. The line y = 0 is called the asymptote of the graph, it represents the value that f ( x ) will never quite reach. We can also say that $\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$ is asymptotic to the line y = 0.
::请注意, 随着 x 的值越来越大, 图形会越来越接近 x 轴。在函数值方面, 我们可以说, 当 x 越来越大, f(x) 越来越接近 0 。形式上, 这种函数的行为被称为限制。我们说, 当 x 接近无穷度时, 函数的极限是 0 。线 y = 0 被称为图的空点, 它代表了 f(x) 永远不会达到的值。我们还可以说, f(x) = 1x 是线 y = 0 的最小值。

If we consider the behavior of the function as x approaches $\begin{aligned} - \infty \end{aligned}$ , we see the same result: the limit of the function has x approaches $\begin{aligned} - \infty \end{aligned}$ is also 0. Notice that this has the same asymptote: y = 0.
::如果我们将函数的行为视为 x 方针 + + ,我们就会看到同样的结果: 函数的极限是 x 方针 + + 也为 0 。请注意, 此函数的默认值相同 : y = 0 。

To be even more formal, we can write limits using a special notation. For the first limit, we write: $\begin{aligned} lim_{x \to \infty} f (x) = 0. \end{aligned}$ For the second limit, we write $\begin{aligned} lim_{x \to - \infty} f (x) = 0. \end{aligned}$ For any limit, here we will always write the x under the abbreviation “lim”, and then we will write the function under consideration. We can also write each of these limits with the specific function:
::更正式地说, 我们可以使用特殊标记来写入限制。对于第一个限制, 我们写 : limxf( x)=0。对于第二个限制, 我们写 limxf( x)=0。对于任何限制, 我们总是在“ lim” 缩略语下写 x , 然后我们写下要考虑的函数。我们也可以用特定的函数来写这些限制 :

$\begin{aligned} lim_{x \to \infty} \frac{1}{x} = 0 \end{aligned}$ and $\begin{aligned} lim_{x \to - \infty} \frac{1}{x} = 0 \end{aligned}$ .
::立方公尺1x=0和立方公尺1x=0。

Because we are focused on end behavior, we are considering the limit of functions as x approaches $\begin{aligned} \pm \infty \end{aligned}$ , and so the asymptotes we will find are horizontal lines. If we were examining other aspects of functions, we might find asymptotes that are vertical lines. For example, the function $\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$ has a vertical asymptote at x = 0, or the y -axis. That is, the graph approaches the y -axis, as x values get closer and closer to 0.
::由于我们专注于最终行为, 我们正在考虑函数的极限, 作为 x 接近 {} {} , 因此我们能找到的微量线是水平线。如果我们正在检查函数的其他方面, 我们可能会找到垂直线的微量线。例如, 函数 f( x) = 1x 在 x = 0 或 Y 轴上有一个垂直的静度。也就是说, 图形接近 y 轴, 因为 x 值越来越接近 0 。

Examples
::实例

Example 1
::例1

Earlier, you were given a question about the distance involved in a strange walk towards a wall.
::早些时候,有人问过你在朝墙走的奇怪路程中距离有多远

If you start 4ft from a wall, and halve the distance to it with each step, how many steps will it take, and how far will you walk, before you actually touch the wall?
::如果你从一堵墙上开始四英尺, 并每一步把距离减半, 它需要多少步骤, 你会走多远, 在你实际触碰墙之前?

Logically, we know that there is only a total distance of 4 feet between you and the wall, so no matter how you break it up, you cannot walk more than 4 feet. However, the actual distance you cover, and the number of steps it would take, cannot truly be defined since there could always be 1/2 of the remaining distance left. Technically speaking, you could continue the process forever without actually touching the wall! Of course, in practice, your ability to only move 1/2 of the remaining distance is limited by muscle control and measurement accuracy, so you would touch the wall before very many steps were actually taken.
::从逻辑上讲,我们知道,你和隔离墙之间只有四英尺的总距离,所以无论你如何分裂,你都无法走超过四英尺。然而,你所覆盖的实际距离和所要采取的步骤,由于所剩距离的长度总是1/2,因此无法真正确定。技术上讲,你可以永远地继续这一进程,而不必实际触动墙。当然,实际上,你仅移动所剩距离的五分之一的能力受到肌肉控制和测量精确度的限制,因此,在采取许多步骤之前,你可以触摸墙。

Mathematically: $\begin{aligned} lim_{n \to \infty} (4 - \frac{4}{2^{n}}) = 4 \end{aligned}$ , where $\begin{aligned} n \end{aligned}$ is the number of steps.
::数学: limn( 4-42n) = 4, 其中 n 是步骤数。

In other words: As the remaining distance gets closer and closer to 0, the total distance approaches 4.
::换句话说,随着剩下的距离越来越接近零,总距离接近4。

Example 2
::例2

Write the limit described using limit notation.
::使用限制符号写入描述的写限制。

The limit of some function $\begin{aligned} f (x) \end{aligned}$ as x approaches infinity is 2.
::某些函数 f(x) 作为 x 方法的无穷限制为 2 。

We write the limit as follows:
::兹将限额写如下:

$\begin{aligned} lim_{x \to \infty} f (x) = 2 \end{aligned}$
::limxf(x)=2

Example 3
::例3

Explain in words the meaning of the limit statement: $\begin{aligned} lim_{x \to \infty} (3 + \frac{2}{x}) = 3 \end{aligned}$
::用文字解释限制语句的含义: limx( 3+2x)=3

$\begin{aligned} lim_{x \to \infty} (3 + \frac{2}{x}) = 3 \end{aligned}$ means: "As larger and larger numbers are substituted in for $\begin{aligned} x \end{aligned}$ in the function $\begin{aligned} 3 + 2 / x \end{aligned}$ , the value comes closer and closer to $\begin{aligned} 3 \end{aligned}$ .
::limx( 3+2x) = 3 表示 : “ 函数 3+2/x 中 x 的数值越大, 值就越接近 3 。

This is due to the fact that the value added to $\begin{aligned} 3 \end{aligned}$ gets smaller and smaller, down to effectively as $\begin{aligned} 2 \end{aligned}$ is divided by larger and larger numbers.
::这是因为增加值为3的增值越来越小,而实际的增值为2除以较大和较大数量。

Example 4
::例4

Determine the horizontal asymptote of the function $\begin{aligned} g (x) = \frac{2 x - 1}{x} \end{aligned}$ and express the asymptotic relationship using limit notation.
::确定函数 g( x) = 2x- 1x 的水平等同度, 使用限制符号表达无ymptatic 关系。

This function is asymptotic to the line y = 2.
::此函数对行y=2是无症状的。

The limit is written as $\begin{aligned} lim_{x \to\neq \infty} \frac{2 x - 1}{x} = 2 \end{aligned}$ .
::限制以 limx=% 2x- 1x=2 写入。

We can determine the asymptote (and hence the limit) if we look at the graph. However, we can also analyze the equation to determine the limit. Consider the function $\begin{aligned} g (x) = \frac{2 x - 1}{x} \end{aligned}$ . As x approaches infinity, the x values are getting larger and larger. For sufficiently large values of x , the values of the expression 2 x - 1 are very close to the values of the expression 2 x , because subtracting one from a large number is fairly insignificant. Thus for sufficiently large values of x, $\begin{aligned} \frac{2 x - 1}{x} \approx \frac{2 x}{x} \approx 2 \end{aligned}$ . As you can see from the accompanying table, which was created by a TI-83 graphing calculator, the function value gets closer to 2 as we look at larger and larger x values.
::如果我们查看图形, 我们可以确定最小值( 并由此确定极限 ) 。但是, 我们也可以分析方程来决定极限。考虑函数 g( x) = 2x - 1x 。随着 x 接近无穷度, x 值越来越大。对于足够大的 x 值, 表达 2x - 1 的值非常接近表达 2x 的值, 因为从大数中减去一个值是相当微不足道的。因此, 足够大的 x, 2x - 1x 2xxxx2xxxx}2. 正如您从由 TI- 83 图形化计算器创建的附加表格中看到的那样, 当我们查看较大和更大的 x 值时, 函数值会接近于 2 。

Example 5
::例5

Describe the following case and sketch a graph of the function with the given properties: $\begin{aligned} lim_{x \to - \infty} f (x) = 0 \end{aligned}$ .
::描述以下的大小写和草图,用给定属性绘制函数图:limxf(x)=0。

This reads: "The limit of f(x) as x approaches negative infinity is 0 ." In other words, as x gets massively negative, f(x) or y gets infinitely close to 0.
::这段文字是:“xx对x的极限是0。 ”换句话说, xx产生大量负数, f(x) orygets 无限接近 0 。

There are a number of possible graphs for this case; one example is offered below.
::这个案例有若干可能的图表;下面举一个例子。

Example 6
::例6

Describe the following case and sketch a graph of the function with the given properties: $\begin{aligned} lim_{x \to 4} f (x) = 3 \end{aligned}$ .
::描述下列案例和草图,用给定属性绘制函数图: limx4f(x)=3。

This reads: "The limit of f(x) as x approaches 4 is 3 ." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely close to 3. This can be a straight line, as y approaches 3 when 4 approaches 4 from either direction.
::这段文字是:“xxxxapproaches4is3” 。换句话说,“xxgets 无限接近 4,f(x) orygets 无限接近 3 。这可能是一条直线, 当4个方向向4个方向接近 4个方向时, Asyoproaches 3 。

There are a number of possible graphs for this case; one example is offered below.
::这个案例有若干可能的图表;下面举一个例子。

Review
::回顾

Define the terms horizontal asymptote and vertical asymptote.
::定义水平单点和垂直单点。

Explain the difference between $\begin{aligned} lim_{x \to - 6} f (x) = \infty \end{aligned}$ and $\begin{aligned} lim_{x \to \infty} f (x) = - 6 \end{aligned}$ .
::解释 limx6f( x) 和 limxf( x) 6 之间的区别。

Explain what $\begin{aligned} lim_{x \to \infty} f (x) = 200 \end{aligned}$ means.
::解释一下 limxf(x)=200 意味着什么。

Explain what $\begin{aligned} lim_{x \to 175} f (x) = 175 \end{aligned}$ means.
::解释一下 limx175f(x)=175意味着什么。

Evaluate the following limits, if they exist. If a limit does not exist, explain why.
::如果存在下列限制,请评估这些限制。如果没有限制,请解释原因。

$\begin{aligned} lim_{t \to \infty} \frac{3 t^{2} - 7 t}{t - 8} \end{aligned}$
::立方公尺3t2 - 7t- 8

$\begin{aligned} lim_{t \to \infty} 3 \end{aligned}$
::limt3

$\begin{aligned} lim_{t \to \infty} (t^{2} - t^{4}) \end{aligned}$
::limt( t2- t4)

$\begin{aligned} lim_{x \to \infty} x + \sqrt{x^{2} + 2 x} \end{aligned}$
::limxx+x2+2x

Find the horizontal and vertical asymptotes of the following function: $\begin{aligned} h (g) = \frac{5 g^{2} - 7 g + 9}{g^{2} - 2 g - 3} \end{aligned}$
::查找以下函数的水平和垂直单位数 : h( g) =5g2 - 7g+9g2 - 2g- 3

Given: $\begin{aligned} f (x) = \frac{x^{2} - x - 6}{x^{2} - 2 x - 8} \end{aligned}$ perform the following:
::具有: f(x) =x2 -x6x2 - 2x-8 表现如下:

Find the horizontal and vertical asymptotes. Determine the behavior of $\begin{aligned} f \end{aligned}$ near the vertical asymptotes.
::查找水平和垂直的静态。确定在垂直静态附近 f 的行为。

Find the roots, y intercept and “holes” in the graph.
::在图中找到根、截取和“洞”。

Determine $\begin{aligned} lim_{t \to \infty} \frac{1}{t^{n}} \end{aligned}$ if:
::确定 limt1tn 是否 :

$\begin{aligned} n > 0 \end{aligned}$
::n>0

$\begin{aligned} n < 0 \end{aligned}$
::n<0

$\begin{aligned} n = 0 \end{aligned}$
::n=0

Let G & H be polynomials. Find $\begin{aligned} lim_{x \to \infty} \frac{G (x)}{H (x)} \end{aligned}$ if:
::让 G & H 成为多边协议。如果 :

The degree of G is less than the degree of H
::G级比H级小

The degree of G is greater than the degree of H
::G级高于H级

The degree of G is the same as the degree of H
::G级与H级相同

A pool contains 8000 L of water. An additive that contains 30g of salt per liter of water is added to the pool at a rate of 25 L per minute. a) Show that the concentration of salt after t minutes in grams per liter is: $\begin{aligned} C (t) = \frac{(t) 30 g \cdot 25}{8000 l + 25 (t) l} \end{aligned}$ b) What happens to the concentration as time increases to $\begin{aligned} \infty \end{aligned}$ ? Physically, why does this make sense?
::水池含8000升水。每升水含30克盐的添加剂以每分钟25升的速度添加到水池中。a) 显示每升水的盐浓度为:C(t) =(t) 30g228000l+25(t)lb) 浓度随着时间增加而增加到 ?从身体上看,这为什么有意义?

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。

Section outline

General

限制和零用量

Limits and Asymptotes ::限制和零用量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Limits and Asymptotes
::限制和零用量

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)