Course: 8.3 无限限制 | The Way To Learn

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无限限制
Geeks-R-Us sells titanium mechanical pencils to computer algorithm designers. In an effort to attract more business, they decide to run a rather unusual promotion:
::Geeks-R-Us向计算机算法设计师出售钛机械铅笔。为了吸引更多的生意,他们决定经营一个非常不寻常的促销:

"SALE!! The more you buy, the more you save! Pencils are now $\begin{align*}\$ \frac{12x}{x - 3}\end{align*}$ per dozen!"
::你买得越多,救得越多!

If the tri llionaire, Spug Dense, comes in and says he wants to buy as many pencils as Geeks-R-Us can turn out, what will the cost of the pencils approach as the order gets bigger and bigger?
::如果万亿富翁Spug Dense 进来说他想买多少铅笔只要Geeks-R-Us能发现, 铅笔的成本会随着订单的越来越高而增加多少?

Infinite Limits
::无限限制

Sometimes, a function may not be defined at a particular number, but as values are input closer and closer to the undefined number, a limit on the output may not exist. For example, for the function f ( x ) = 1/x (shown in the figures below), as x values are taken closer and closer to 0 from the right, the function increases indefinitely. Also, as x values are taken closer and closer to 0 from the left, the function decreases indefinitely.
::有时,函数可能无法在一个特定数字上定义,但由于数值是输入接近和接近未定义数字的输入,输出可能不存在限制。例如,对于函数 f(x) = 1/x(在下图中显示),当x 值从右侧接近和接近0 时,函数将无限期增加。此外,由于x 值从左侧接近和接近0,函数将无限期减少。

We describe these limiting behaviors by writing
::我们用书面描述这些限制行为。

$\begin{align*}\lim_{x \rightarrow 0^+} \frac{1} {x} = + \infty\end{align*}$
::立方公尺x0+1x

$\begin{align*}\lim_{x \rightarrow 0^-} \frac{1} {x} = - \infty\end{align*}$
::立方公尺

Sometimes we want to know the behavior of f ( x ) as x increases or decreases without bound. In this case we are interested in the end behavior of the function, a concept you have likely explored before. For example, what is the value of f ( x ) = 1/x as x increases or decreases without bound? That is,
::有时我们想要知道 f( x) 作为 x 递增或递减而不受约束的行为。在此情况下, 我们对函数的最终行为感兴趣, 您以前可能已经探索过这个概念。例如, f( x) = 1/x 作为 x 递增或递减而不受约束的值是多少 ? 也就是说, f( x) = 1/x 作为 x 递增或递减是多少 ?

$\begin{align*}\lim_{x \to +\infty} \frac{1} {x} = ?\end{align*}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么?

$\begin{align*}\lim_{x \to -\infty} \frac{1} {x} = ?\end{align*}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么?

As you can see from the graphs (shown below), as x decreases without bound, the values of f ( x ) = 1/x are negative and get closer and closer to 0. On the other hand, as x increases without bound, the values of f ( x ) = 1/x are positive and still get closer and closer to 0.
::从图表(如下表所示)可以看出,当x无约束地下降时,f(x)=1/x的值为负值,接近于0。另一方面,当x无约束地增加时,f(x)=1/x的值为正值,仍然接近于0。

That is,
::也就是说,

$\begin{align*}\lim_{x \to +\infty} \frac{1} {x} = 0\end{align*}$
::limx1x=0

$\begin{align*}\lim_{x \to -\infty} \frac{1} {x} = 0\end{align*}$
::limx1x=0

Examples
::实例

Example 1
::例1

Earlier, you were asked a question about buying a lot of pencils.
::之前有人问过你买很多铅笔的问题

As Spug buys more and more pencils, the cost of each dozen will drop quickly at first, and level out after a while, approaching $12 per dozen.
::斯普格购买的铅笔越来越多, 每12支铅笔的成本一开始会迅速下降, 过一阵子就会稳定下来, 接近每12美元12美元。

You can see the effect on the graph here:
::您可以在此看到图中的效果 :

Example 2
::例2

Evaluate the limit by making a graph: $\begin{align*}\lim_{x\to 3^+} \frac{x + 6}{x - 3}\end{align*}$ .
::通过绘制一个图: limx3+x+6x-3, 来评估极限值。

By looking at the graph:
::通过查看图表:

We can see that as x gets closer and closer to 3 from the positive side, the output increases right out the top of the image, on its way to $\begin{align*}\infty\end{align*}$
::我们可以看到,当x从正面接近3时, 输出会从图像的顶部上升,

Example 3
::例3

Evaluate the limit: $\begin{align*}\lim_{x\to \infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 3}\end{align*}$ .
::评估极限值: limx11x3- 14x2+8x+169x-3。

To evaluate , a little bit of intuition helps. Let's think this one through.
::评估一下,直觉能帮上点忙。让我们好好想想这个。

First, note that since we are looking at what happens as $\begin{align*}x \to \infty\end{align*}$ most of the interesting stuff will happen as x gets really big.
::首先,请注意,既然我们是在看什么发生作为x 大部分有趣的事情会发生当x变得真正大。

On the top part of the fraction, as x gets truly massive, the 11x ³ part will get bigger much faster than either of the other terms. In fact, it increases so much faster than the other terms completely cease to matter at all once x gets really monstrous. That means that the important part of the top of the fraction is just the 11x ³ .
::在分数的顶端, 当 x 变得真正巨大时, 11x3 部分会比其他两个条件中的任何一个大得多。事实上, 它的增加比其他条件都快得多。一旦 x 变得非常可怕, 它就会完全停止重要。这意味着该分数顶部的重要部分只是 11x3 。

On the bottom, a similar situation develops. As x gets really, really big, the -3 matters less and less. So the bottom may as well be just 9x .
::在底部, 类似的情况也会发展。当 x 变大时, - 3 的值越来越小。所以底部的值可能只有 9x 。

That gives us $\begin{align*}\frac{11x^3}{9x}\end{align*}$ which reduces to $\begin{align*}\frac{11x^2}{9}\end{align*}$
::这让我们11x39x 减到11x29

Now we can more easily see what happens at the "ends." As x gets bigger and bigger, the numerator continues to get bigger faster than the denominator, so the overall output also increases.
::现在,我们可以更容易地看到“端”会发生什么。随着 x 越来越大, 分子继续变得更快于分母, 所以总输出也会增加。

$\begin{align*}\therefore \lim_{x\to +\infty} \frac{11x^3 - 14x^2 +8x +16} {9x - 3}\text{ is }+\infty\end{align*}$
::============================================================================================================================================== ================================================================================================================================================================

Example 4
::例4

Evaluate $\begin{align*}\lim_{x\rightarrow 0} \frac{x + 2} {x + 3} \end{align*}$ .
::评估 limx=0x+2x+3。

This one is easier than it looks! As x --> 0, it disappears, leaving just the fraction: 2/3
::这个比看起来容易得多。作为 x - > 0, 它会消失, 只留下分数: 2/3

Example 5
::例5

Make a graph to evaluate the limit $\begin{align*}\lim_{x\rightarrow \infty} \frac {1} {\sqrt{x}} \end{align*}$ and $\begin{align*}\lim_{x\rightarrow 0^{+}} \frac {1} {\sqrt{x}} \end{align*}$ .
::绘制一个图表以评价 limx1x 和 limx0+1x 的限值。

By looking at the image, we see that as x gets huge, so does $\begin{align*}\sqrt{x}\end{align*}$ which means that 1 is being divided by an ever-larger number, and the result is getting smaller and smaller.
::通过观察图像,我们可以看到,当x变大时,x值也变大,x值也变大,这意味着1值被一个越来越大的数字除以,结果越来越小。

$\begin{align*}\lim_{x\rightarrow \infty} \frac {1} {\sqrt{x}} \end{align*}$ = 0
::1x=0

On the same image, we can see that as x gets closer and closer to zero, so does $\begin{align*}\sqrt{x}\end{align*}$ which means that 1 is being divided by an ever smaller number, and the result gets bigger and bigger.
::在同一图像上,我们可以看到,当x接近零时,x接近零时,“x”也接近零时,“x”也接近零时,这意味着1除以一个越来越小的数字,结果越来越大。

$\begin{align*}\lim_{x\rightarrow 0^{+}} \frac {1} {\sqrt{x}} \end{align*}$ is $\begin{align*}+\infty\end{align*}$
::立方公尺======================================================================================================================================================================================================================

Example 6
::例6

Graph and evaluate the limit: $\begin{align*}\lim_{x\to 2^{+}} \frac {1} {x - 2}\end{align*}$ .
::图表和评估极限: limx%2+1x-2。

By looking at the image, we can see that as x gets closer and closer to 2 from the positive direction, 1 gets divided by smaller and smaller numbers, so the result gets larger and larger.
::通过观察图像,我们可以看到,当x从正向接近2时,1时除以较小数目,结果就会越来越大。

$\begin{align*}\lim_{x\to 2^{+}} \frac {1} {x - 2}\text{ is }+\infty\end{align*}$
::limx% 2+1x-2 是 @ @ label

Review
::回顾

Evaluate the limits:
::评估限度:

$\begin{align*}\lim_{x\to 3^{-}} \frac {1} {x - 3}\end{align*}$
::立方厘米 3 - 1x - 3

$\begin{align*}\lim_{x\to -4^{+}} \frac {1} {x + 4}\end{align*}$
::limx4+1x+4

$\begin{align*}\lim_{x\to -\left(\frac{8}{3}\right)^{+}} \frac {1} {3x + 8}\end{align*}$
::limx(83)+13x+8

$\begin{align*}\lim_{x\to -5^{+}} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}$
::limx5+(x2+11x+30x+30x+5)

$\begin{align*}\lim_{x\to -\infty} \frac{\left(x^2+11x+30\right)}{x+5}\end{align*}$
::limx(x2+11x+30x+30x+5)

$\begin{align*}\lim_{x \to \infty}\end{align*}$ $\begin{align*}\frac{-11x^3 + 20x^2 + 15x - 17}{-9x^3 + 5x^2 - x - 17}\end{align*}$
::立方公尺xxx3+20x2+15x-7-9x3+5x2-x-17

$\begin{align*}\lim_{x \to \infty} 13 \end{align*}$
::立方公尺13

$\begin{align*}\lim_{x \to \infty} \frac{-2x + 18}{17x - 3}\end{align*}$
::limx% 2x+1817x- 3

$\begin{align*}\lim_{x \to \infty} 15\end{align*}$
::limx% 15

$\begin{align*}\lim_{x \to \infty} -5x^2 + 5x + 14\end{align*}$
::limx=%5x2+5x+14

$\begin{align*}\lim_{x \to \infty}7x + 12\end{align*}$
::limx=% 7x+12

$\begin{align*}\lim_{x \to \infty} -3x + 13\end{align*}$
::limx=%3x+13

$\begin{align*}\lim_{x \to \infty} \frac{13x - 8}{19x^3 - 11x^2 + x + 4}\end{align*}$
::立方公尺13x-819x3-11x2+x+4

$\begin{align*}\lim_{x \to \infty} -17x + 14\end{align*}$
::limx17x+14

$\begin{align*}\lim_{x \to \infty}-7x^2 - 2x - 13\end{align*}$
::立方 7x2 - 2x - 13

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

无限限制

Infinite Limits ::无限限制

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)

Infinite Limits
::无限限制

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)