从表中估计限制

限制告诉我们很多关于函数行为的信息。它们帮助数学家和工程师对函数、行为和属性进行推理。它们构成了微积分中几乎所有重要概念的基础。限制帮助我们估计值函数似乎在特定点上采用。通常这些限制很简单并且可以很容易地计算出来，但有时这些限制会评估为某种不确定的形式，为了解决它们有很多技术和技巧。使用表格求解极限是适用于几乎所有类型极限的方法之一。

限制

极限是函数在特定点明显采用的值。对于定义在实数上的特定函数f(x)，函数在任意点 x = c 处的极限表示为$\lim_{x \to c}f(x)$ .请记住，限制不是函数在特定点的值，限制表示当一个人接近该点时函数似乎正在采用的值。通常，从点的左侧和右侧都接近极限。

通常限制可以通过简单的替换来计算，但有时某些限制采用一些未定义的形式。这些形式被称为不可确定的形式。不确定形式的例子是$\frac{\infty}{\infty}$ , $\frac{0}{0}$ , 0 x ∞ 等。

使用表格可以避免所有这些问题和表格。让我们详细看看。

使用表格逼近极限

表格是一个很好的工具，可以用来推断限制。使用表格还可以处理无法确定的限制形式。该方法涉及在我们应该计算极限的点附近的点处计算函数的值。以这种方式估计极限比目测函数图要好得多。让我们考虑一个示例，通过采用函数f(x) = x – 1 并计算此函数在 x = 0 处的极限来了解此方法的工作原理。

$\lim_{x \to 0}x - 1 = -1$

让我们使用表格来计算这个值。方法是计算函数在不同x值处的值。目标是无限接近目标点而不是目标点。请记住，在创建表格时，应从左侧和右侧访问该函数。

通过表格计算限制的步骤：

1. Start from the points which are infinitely close to the target point.
2. Approach the target point from left and go infinitely close while evaluating the function for every point.
3. Repeat the same thing from right-hand side.

The table that is created will have values that are almost equal, and will give an estimate of the limit at that point. 

x	f(x)
-0.1	-1.1
-0.05	-1.05
-0.001	-1.001
0.001	0.999
0.05	0.95

请注意，在表格中，随着我们从任一侧越来越接近 x = 0，函数的值接近值 -1。

因此， $\lim_{x \to 0}x - 1 = -1$

Note: While populating the table, some things must be kept in mind to get the right value of limit: 

1. Do not assume that the function value is the value of the limit. Sometimes there might be a discontinuity in the function, it might seem that function is going to take a particular value, but the actual value at that point is different. It often happens at the points where the function is either discontinuous or undefined.
2. Approach from both sides of the point.
3. Always go as close as possible to the point.

表格的单边限制

虽然要求提供单边限制，但表格中通常填充的值大于和小于要计算限制的点。换句话说，要计算左侧和右侧极限。在单边限制中，表格由点的左侧或右侧填充。

让我们通过一个例子来看看。

示例：考虑一个示例，对于函数f(x) = x ² 。计算$\lim_{x \to 0^{+}}f(x)$

解决方案：

$\lim_{x \to 0^{+}}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 0. 

x	f(x)
0.1	0.01
0.05	0.0025
0.001	0.000001
0.0005	0.00000025
0.0001	0.00000001

Notice from the table that the value of the limit is approaching the value 0. 

$\lim_{x \to 0^{+}}f(x) = 0$

让我们看看这个概念的一些问题。

示例问题

问题 1：考虑一个例子，对于函数f(x) = x ²计算$\lim_{x \to 1}f(x)$

解决方案：

$\lim_{x \to 1}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 1. 

x	f(x)
0.9	0.81
0.95	0.9025
0.99	0.9801
0.999	0.99801
1.001	1.002
1.01	1.02

Notice from the table that the value of the limit is approaching the value 1. 

$\lim_{x \to 0}f(x) = 1$

问题 2：考虑一个例子，函数f(x) = 5x。计算$\lim_{x \to 0}f(x)$

解决方案：

$\lim_{x \to 0}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 0. 

x	f(x)
-0.1	-0.5
-0.05	-0.25
-0.001	-0.005
0.001	0.005
0.01	0.05

Notice from the table that the value of the limit is approaching the value 0. 

$\lim_{x \to 0}f(x) = 0$

问题 3：考虑一个例子，对于函数f(x) = $\frac{x^2 - 1}{x - 1}$ .计算$\lim_{x \to 1}f(x)$

解决方案：

$\lim_{x \to 1}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 1. 

x	f(x)
0.999	1.999
0.9999	1.9999
0.99999	1.99999
1.00001	2.00001
1.0001	2.0001

Notice from the table that the value of the limit is approaching the value 2. 

$\lim_{x \to 1}f(x) = 2$

问题 4：考虑一个例子，函数f(x) = $\frac{1}{x + 1}$ .计算$\lim_{x \to 2}f(x)$

解决方案：

$\lim_{x \to 2}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 2. 

x	f(x)
1.999	0.33344448
1.9999	0.33334444
1.99999	0.33333444
2.00001	0.33333222
2.0001	0.33332222

Notice from the table that the value of the limit is approaching the value 0.333. 

$\lim_{x \to 1}f(x) = 0.333$

问题 5：考虑一个例子，对于函数f(x) = $\frac{sin(x)}{x}$ .计算$\lim_{x \to 0}f(x)$

解决方案：

$\lim_{x \to 0}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 0. 

x	f(x)
-0.1	0.99833
-0.01	0.99998
-0.001	0.99999
0.001	0.99999
0.01	0.99998

Notice from the table that the value of the limit is approaching the value 1. 

$\lim_{x \to 1}f(x) = 1$

问题 6：考虑一个例子，函数f(x) = |x|。计算$\lim_{x \to 0}f(x)$

解决方案：

$\lim_{x \to 0}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 0. 

x	f(x)
-0.1	0.1
-0.01	0.01
-0.001	0.001
0.001	0.001
0.01	0.01

Notice from the table that the value of the limit is approaching the value 0. 

$\lim_{x \to 1}f(x) = 0$

问题 7：考虑一个例子，函数f(x) = log(x)。计算$\lim_{x \to 1}f(x)$

解决方案：

$\lim_{x \to 1}f(x)$

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 1. 

x	f(x)
0.999	0.000434
0.9999	0.0000434
0.99999	0.00000434
1.00001	0.00000434
1.0001	0.0000434

Notice from the table that the value of the limit is approaching the value 0. 

$\lim_{x \to 1}f(x) = 0$