- Calculus & It’s Application- Chapter 2 Introduction to Limits 朝陽科技大學 資訊管理系 李麗華 教授

2 Introduction to Limits Since 16 century, there were many scientists used calculus-like method to solve problems. However, the inventing of calculus should be credited to Newton and Leibniz. Newton invented calculus in 1665, however he took more than 20 years to publish his results. Leibniz was thought as the first man who published the theory of calculus and the notation he proposed has been used until today. Calculus application has found in almost every areas, such as speed of blood flows, size of packaging, maximization of the product level.

3 Introduction to Limits In this chapter, we will introduce the following topics: –Introduction to limits –Continuity –One-sided limits –Limits at infinity –Infinite limits

4 2-1 Introduction to Limits 極限 (Limits) 在日常生活中極常出現， 例如 : 繩子每天取走現有長度一半 (1, 1/2, 1/4, 1/8, 1/16…..) 終有一天會取到接近於無 ( 即 0) ， 所以我們會說，剩餘長度幾乎為 0 。 Q: 認真想一下 …. 真的會為 0 嗎 ?

5 2-1 Introduction to Limits [ 例一 ] 有下列函數 ， 由此函數知， 不可為 1 若代入接近 1 的數，可發現 : 可觀察到 當 故我們可說，當 趨近 1 時， 趨近 4 ，其寫法為：

6 2-1 Introduction to Limits [ 例二 ] 若一火車每小時時速 60 mile/hr ， 若 趨近於 3 小時，請問距離將趨近多少 ?

7 2-1 Introduction to Limits 定義 –For any function, means that as gets closer and closer to, Gets closer and closer to –Limit Theorems: If, and are real numbers, then C 為常數

8 2-1 Introduction to Limits 練習 :

9 2-1 Introduction to Limits 上台練習 :

Continuity 生活中常見連續的事件，在數學中連續 (Continuity) 則指函數圖形在 處不中斷，以 下用四個圖來說明 當時不連續 未定義 當時不連續 不存在 1.2.

Continuity 當時不連續 3. 當 4. 時 由上可知，圖 1 、 2 、 3 在 時均為斷裂的， 只有圖 4 連續。

Continuity If the graph of a function has no break, gap, holes, or jumps in them, then this function can be called as a continuous function If is continuous at, then If then is continuous at. 本定義包含三項要素，即 (1) 一定要存在， (2) 必有定義 ( 值 ) ， (3)

Continuity 例一 : ，若 ，則 當 時連續 例二 : ，當 時 當 時連續 例三 : ，由本函數知 但 沒有定義，故 不連續

Continuity 上台練習 : – 畫圖並找 連續或不連續