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1、英文原文 An Intuitive Introduction to Limits Limits is one of the most fundamental concepts of calculus. The concept of calculus was not that clear during the time of Leibniz and Newton, but later developments on the concept, particularly the definition by Cauchy, Weierstrass and other mathematicians es
2、tablished the firm foundation of calculus. In the discussion below, I will try to introduce the concept of limits intuitively as it appears in common problems. Circumference and Limits If we are going to approximate the circumference of a circle using the perimeter of an inscribed polygon, even with
3、out computation, we can observe that as the number of sides of the polygon increases, the better our approximation . In fact, we can make the perimeter of the polygon as close as we please to the circumference of the circle by choosing a sufficiently large number of sides. Moreover, no matter how la
4、rge the number of sides our polygon has, its perimeter will never exceed the circumference of the circle. As the number of side of the polygons increases, its perimeter gets closer to the circumference of the circle. In a more technical term, we say that the limit of the perimeter of the polygon as
5、the number of its sides increases without bound is equal to the circumference of the circle. Symbolically, if we let be the number of sides of the inscribed polygon, be the perimeter of a polygon with sides, and be the circumference of the circle, we can say that the limit of as is equal to . Compac
6、tly, we can write .Numbers and Limits We end with a more familiar example usually found in books. What if we want to find the limit of as approaches 3? To answer the question, we must find the value where is very close to 3. Those values would be numbers that are very close to 3some slightly greater
7、 than 3 and some slightly less than 3. Place the values in a table we have From the table, we can clearly see that as the value of approaches 3, the value of approaches 7. Concisely, we can write the . 英文翻译极限的一些直观介绍 极限是一个最基本的概念的演算。微积分的概念在牛顿莱布尼兹时代并不明确，直到后面柯西、魏尔斯特拉斯和其他数学家的定义才奠定了牢固的基础微积分。在下面的讨论中，我将试图引入
8、概念出现在普通问题中的直观极限。圆和极限 如果我们要使用多边形近似的计算圆的周长，即使没有计算时，我们能观察到，随着多边形边长数量的增加，它越来越接近圆。事实上，我们能使多边形尽量接近圆的圆周来计算圆的周长。此外，无论有多大的多边形数量，它们永远不会超过周长。. 作为多边形数量方面的增加,它的周长愈接近的圆周圆 更技术性的术语，我们说的多边形边的数量无限接近圆周时，多边形和圆的面积是几乎相等的。一般的，我们设多边形的边数为， 为多边形的周长，为圆的周长，我们可以说，当时的极限是.简洁地,我们能写成.值和极限 我们通常可以在书中找到一个更为熟悉的例子。我们怎样求出当趋于3时的极限? 要回答这个问题,我们必须找到无限趋近3时的值。这些值必须是那些非常接近3（一些比3大一点点和一些比3小一点点）的数的值。下表列出了一些数的值从表中可以看出,我们可以清楚地看到当趋于3时的值趋近7.简要的,我们可以写成。
