极限一些直观介绍外文翻译.doc

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 established 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 without 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 large 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 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 . Compactly, 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 3–some slightly greater 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