Section: 限制属性 | 2.3 限制的属性

Section outline

You're familiar with the idea of a limit of a function, and that some limits a re computed using numerical and graphical methods. Limits can also be evaluated using the properties of limits. How would you find $\begin{aligned} lim_{x \to 0} \frac{x^{2} + 5 x}{x} \end{aligned}$ without using a graph or using a table of values?

Properties of Limits
::限制属性

Let’s begin with some observations about limits of some simple functions. Consider the following limit problems: $\begin{aligned} lim_{x \to 2} 5 \end{aligned}$ and $\begin{aligned} lim_{x \to 4} x \end{aligned}$ .
::让我们先观察一下某些简单功能的局限性。考虑一下以下的限制问题: limx%25 和 limx%4x 。

We note that each of these functions is defined for all real numbers. If we apply our intuition for finding the limits we conclude correctly that: $\begin{aligned} lim_{x \to 2} 5 = 5 \end{aligned}$ and $\begin{aligned} lim_{x \to 4} x = 4 \end{aligned}$ .
::我们注意到,这些功能中的每一功能都是为所有实际数字所定义的。如果我们运用直觉来找到极限,我们就会正确地得出结论:limx25=5和limx4x=4。

The above results can be encapsulated in the following limit properties:
::上述结果可概括于以下限制特性:

Basic Limit Properties:
::基本限制属性 :

$\begin{aligned} lim_{x \to a} c = c \end{aligned}$
::立方厘米=c

$\begin{aligned} lim_{x \to a} x = a \end{aligned}$
::立方厘米=a

Many functions can be expressed as the sums, differences, products, quotients, powers and roots of other more simple functions. The following properties are also useful in evaluating limits:
::许多功能可以表现为其他更简单功能的总和、差异、产品、商数、权力和根基。

More Basic Limit Properties:
::更多基本限制属性 :

If $\begin{aligned} lim_{x \to a} f (x) \end{aligned}$ and $\begin{aligned} lim_{x \to a} g (x) \end{aligned}$ both exist, then
::如果存在 limxaf(x) 和 limxag(x) , 那么

$\begin{aligned} lim_{x \to a} [c f (x)] = c lim_{x \to a} f (x) \end{aligned}$ where $\begin{aligned} c \end{aligned}$ is a real number,
::c 是真实数字的 c=climx=climx=xxxxxxxxxxx

$\begin{aligned} lim_{x \to a} [f (x) \pm g (x)] = lim_{x \to a} f (x) \pm lim_{x \to a} g (x) \end{aligned}$ ,
:xxxxxxxxxxxxxxxx)= limx*5xxxxxx=xxxxxxx=ag(x)x,

$\begin{aligned} lim_{x \to a} [f (x) \cdot g (x)] = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x) \end{aligned}$ ,
:xxxxxxxxxxxx) = limx*5xxxxxxxxxxxxxx)

$\begin{aligned} lim_{x \to a} [\frac{f (x)}{g (x)}] = \frac{lim_{x \to a} f (x)}{lim_{x \to a} g (x)} \end{aligned}$ provided that $\begin{aligned} lim_{x \to a} g (x) \neq 0. \end{aligned}$
:xx)g(x)=limx)af(x)limx=ag(x)x(x)

$\begin{aligned} lim_{x \to a} [f (x)]^{n} = [lim_{x \to a} f (x)]^{n} \end{aligned}$ where $\begin{aligned} n \end{aligned}$ is a real number
::n 是真实数字的 n = [limx_a[f(x)]n= [limx_af(x)]n

$\begin{aligned} lim_{x \to a} \sqrt[n]{f (x)} = \sqrt[n]{lim_{x \to a} f (x)} \end{aligned}$ , where $\begin{aligned} n \end{aligned}$ is either an odd integer or $\begin{aligned} n \end{aligned}$ is an even positive integer and $\begin{aligned} lim_{x \to a} f (x) > 0. \end{aligned}$
::limxaf(x)n =limxaf(x)n,其中n为奇数整数或n为正数整数,而limxaf(x)>0。

Knowing these properties, allows evaluation of the limits of a wide range of functions.
::了解这些特性,就能够评估各种职能的限度。

Take the problem: $\begin{aligned} lim_{x \to 2} (3 x + 7) \end{aligned}$ . Based on the properties above, the limit can be evaluated in the following steps:
::根据上述属性,可按以下步骤评估限额:

$\begin{aligned} lim_{x \to 2} (3 x + 7) & = lim_{x \to 2} (3 x) + lim_{x \to 2} 7 \\ = 3 \cdot lim_{x \to 2} (x) + 7 \\ = 3 \cdot 2 + 7 \\ = 13 \end{aligned}$

::立方公尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺= 立方尺=

Therefore: $\begin{aligned} lim_{x \to 2} (3 x + 7) = 13 \end{aligned}$
::因此: limx%2( 3x+7)=13

Note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of $\begin{aligned} x = 2 \end{aligned}$ in the function.
::请注意,在这种情况下,适用基本限值的属性所产生的限值与函数中的 x=2 的直接替代值相同。

Now, evaluate $\begin{aligned} lim_{x \to 2} (x^{3} \cdot \sqrt{2 x}) \end{aligned}$ .
::现在, 评估 limx_ 2( x3_ 2x) 。

Based on the properties of limits , the limit can be evaluated in the following steps:
::根据限制的特性,可以采取以下步骤评价限制:

$\begin{aligned} lim_{x \to 2} (x^{3} \cdot \sqrt{2 x}) & = lim_{x \to 2} x^{3} \cdot lim_{x \to 2} \sqrt{2 x} \\ = (lim_{x \to 2} x)^{3} \cdot \sqrt{lim_{x \to 2} 2 x} \\ = (2)^{3} \cdot \sqrt{4} \\ = 16 \end{aligned}$

::立方公尺=( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方公尺) =( 立方尺)

Therefore: $\begin{aligned} lim_{x \to 2} (x^{3} \cdot \sqrt{2 x}) = 16 \end{aligned}$
::因此: limx%2(x3%2x)=16

Again, note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of $\begin{aligned} x = 2 \end{aligned}$ in the function.
::请注意,在此情况下,适用基本限制属性的结果是,其限值与函数中的 x=2 的直接替代值相同。

Examples
::实例

Example 1
::例1

$\begin{aligned} lim_{x \to 0} \frac{x^{2} + 5 x}{x} \end{aligned}$

Example 2
::例2

Evaluate $\begin{aligned} lim_{x \to 0} f (x) \end{aligned}$ where $\begin{aligned} f (x) \end{aligned}$ is the rational function $\begin{aligned} f (x) = \frac{x^{3} + 3 x^{2} - 10 x - 24}{x^{2} + 1} . \end{aligned}$
::以 f( x) 为合理函数 f( x) =x3+3x2 - 10x- 24x2+1 的 f( x) 值评价 limx% 0f( x) 。

$\begin{aligned} lim_{x \to 0} f (x) & = lim_{x \to 0} [\frac{x^{3} + 3 x^{2} - 10 x - 24}{x^{2} + 1}] \\ = [\frac{lim_{x \to 0} (x^{3} + 3 x^{2} - 10 x - 24)}{lim_{x \to 0} (x^{2} + 1)}] \\ = [\frac{lim_{x \to 0} x^{3} + lim_{x \to 0} 3 x^{2} - lim_{x \to 0} 10 x - lim_{x \to 0} 24}{lim_{x \to 0} x^{2} + lim_{x \to 0} 1}] \\ lim_{x \to 0} f (x) & = lim_{x \to 0} [\frac{x^{3} + 3 x^{2} - 10 x - 24}{x^{2} + 1}] \\ = [\frac{0 + 0 - 0 - 24}{0 + 1}] \\ = - 24 \end{aligned}$

::==[limx=0x0x0x0x0x0x0x0x0x0x3+3x2-10x-24x2+1x1]=[limx0x0(x3+3x2-10x-2--24x-24x-24x-x2+1)]=[limx=0x0x0x0x0x0x0-024x1]

Example 3
::例3

Find the following limit if it exists: $\begin{aligned} lim_{x \to - 4} \frac{2 x^{2}}{\sqrt{4 - x}} \end{aligned}$
::如果存在以下限制, 则查找该限制 : limx42x24- x

Let’s apply the basic quotient rule to evaluate this limit.
::让我们运用基本商数规则来评估这一限度。

$\begin{aligned} lim_{x \to - 4} \frac{2 x^{2}}{\sqrt{4 - x}} & = \frac{lim_{x \to - 4} 2 x^{2}}{lim_{x \to - 4} \sqrt{4 - x}} \\ = \frac{32}{\sqrt{8}} \\ = 8 \sqrt{2} \end{aligned}$

::==82 =82 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Therefore: $\begin{aligned} lim_{x \to - 4} \frac{2 x^{2}}{\sqrt{4 - x}} = 8 \sqrt{2} \end{aligned}$
::因此: limx*42x24-x=82

Again, the limit result is the same as using direct substitution of $\begin{aligned} x = - 4 \end{aligned}$ in the function.
::同样,限制结果与在函数中使用 x4 的直接替换相同。

Review
::回顾

What is the limit of $\begin{aligned} 3 x^{2} + 4 x - 9 \end{aligned}$ as $\begin{aligned} x \to - 2 \end{aligned}$ ?
::3x2+4x- 9 作为 x2 的上限是多少 ?

What is the limit of $\begin{aligned} 5^{x} \cdot \cos (π x) \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ ?
::5xxxxcos(xx) 作为 x%0 的上限是多少?

What is the limit of $\begin{aligned} 5^{x} + \cos (π x) \end{aligned}$ as $\begin{aligned} x \to 1 \end{aligned}$ ?
::5x+cos(xx) 作为 x% 1 的上限是多少 ?

If the limit of $\begin{aligned} f (x) \end{aligned}$ as $\begin{aligned} x \to 1 \end{aligned}$ is zero, is the limit of $\begin{aligned} f (x) g (x) \end{aligned}$ as $\begin{aligned} x \to 1 \end{aligned}$ always equal to zero, for all $\begin{aligned} g (x) \end{aligned}$ ?
::如果f(x) 的限值为 x% 1 零,那么对于所有 g(x) 来说, f(x) g(x) 的限值是否始终等于 0 ?

What is $\begin{aligned} lim_{x \to 3} \frac{x^{2} + 7 x + 12}{x + 3} \end{aligned}$ ?
::什么是limx=3x2+7x+12x+3?

What is $\begin{aligned} lim_{x \to 2} \sqrt[3]{x^{3} + 4 x^{2} + 3} \end{aligned}$
::什么是 limx% 2x3+4x2+33

Find $\begin{aligned} lim_{x \to - 2} (x + 4)^{3} (2 x - 1)^{2} \end{aligned}$
::查找 limx%2( x+4) 3( 2x- 1) 2

$\begin{aligned} lim_{x \to - 7} \frac{x^{2} + 7 x + 12}{x + 7} \end{aligned}$
::limx=7x2+7x+12x+7

If the limit of $\begin{aligned} f (x) \end{aligned}$ as $\begin{aligned} x \to 5 \end{aligned}$ is one and the limit of $\begin{aligned} g (x) \end{aligned}$ as $\begin{aligned} x \to 5 \end{aligned}$ is two, and both $\begin{aligned} f (x) \end{aligned}$ and $\begin{aligned} g (x) \end{aligned}$ are continuous functions, is it necessarily true that $\begin{aligned} f (5) g (5) = 2 \end{aligned}$ ?
::如果 f( x) 5 的上限为 1, g( x) 5 的上限为 2, f( x) 和 g( x) 是连续函数, 那么 f(5) g(5)=2 是否必然属实 ?

If the limit of $\begin{aligned} f (x) \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ is 5, and $\begin{aligned} g (x) = x^{2} + x \end{aligned}$ , what is the limit of $\begin{aligned} g (f (x)) \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ ?
::如果 f( x) 以 x_0 表示的 f( x) 限值为 5 , g( x) = x2+x, 则 g( f( x) ) 以 x_0 表示的限值是多少 ?

What is the limit of $\begin{aligned} 5 e^{x} + 2 π^{x} + \frac{x + 2}{\cos (x)} \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ ?
::5ex+2x+x+2cos(x) 作为 x%0 的 5ex+2x+x+2cos(x) 限值是多少?

Let $\begin{aligned} h (x) \end{aligned}$ be a function with a finite limit defined everywhere. What is the limit of $\begin{aligned} \frac{3 \sin (x) h (x)}{x^{9} + 8 x^{5} + 10 x^{4} + 9 x + 1} + e^{x} (\cos (h (x) \sin (x + 2 π))) \end{aligned}$ as $\begin{aligned} x \to π \end{aligned}$ ?
::Let h( x) 是一个函数, 其限制范围在任何地方都有定义。 3sin {( x) h( x) x9+8x5+10x4+9x4+9x1+ex( cos) (h( x)sin}(x+2}) 作为 x} 的限度是多少 ?

Say that $\begin{aligned} p (x) = x^{3} + x^{2} + x + 1 \end{aligned}$ . What must be the limit of $\begin{aligned} f (x) \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ if $\begin{aligned} lim p (f (x)) = 0 \end{aligned}$ as $\begin{aligned} x \to 0 \end{aligned}$ ?
::说 p( x) =x3+x2+x+1。如果瘸( f( x)) =0 = x_0, f( x) x2+x1 的极限必须是 x_0 多少 ?

Which limits of $\begin{aligned} \frac{\sin (x)}{x} \end{aligned}$ can you not take with the quotient rule?
::罪(xx)x的哪些限度不能与商数规则相提并论?

Show that $\begin{aligned} lim f (x)^{5} \end{aligned}$ is equal to $\begin{aligned} (lim f (x))^{5} \end{aligned}$ wherever a limit exists without using the power rule. Is this method enough to prove the power rule?
::显示 limf( x) 5 等于 (limf( x) ) 5 , 只要存在限制而不使用权力规则。这个方法是否足以证明权力规则 ?

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

Properties of Limits ::限制属性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Properties of Limits
::限制属性

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)