Calculus Volume 1 · 免费预览
极限的精确定义
ε-δ 定义使极限概念精确化：对任意容差 ε，都存在一个邻域 δ，使函数值保持在容差范围内；同一模板也用来定义连续性、导数和积分。
1. From intuition to precision
From intuition to precision
2. Verifying simple limits
Verifying simple limits
3. Delta as a function of epsilon
Delta as a function of epsilon
4. The definition's descendants
The definition's descendants
5. Reading the definition correctly
In the epsilon-delta definition of limit, which quantity is chosen first and which may depend on it?
6. Why exclude the point
Why does the definition impose 0 less than |x - a| rather than |x - a| less than delta?
7. Linear verification
To prove lim x to 3 of (2x + 1) equals 7 by the definition, we need |f(x) - 7| less than epsilon. What choice of delta works?
8. Bounding the wild factor
Verifying lim x to 2 of x^2 = 4 requires bounding |x + 2|. Why precommit to delta at most 1 before choosing the final delta?
9. Failure for 1/x at 0
Why does the epsilon-delta definition certify that 1/x has no limit as x approaches 0?
10. Quantifier order
The definition says for every epsilon there exists a delta. What happens to the definition if the quantifiers were reversed, for every delta there exists an epsilon?
11. Definition name
The precise definition of limit is known by the names of the two Greek letters it uses. What is it called, in lowercase with a hyphen? Answer like alpha-beta.
12. Output tolerance band
In the definition, the condition |f(x) - L| less than epsilon says the outputs lie within epsilon of L. What single Greek letter names that tolerance? Answer with the letter name spelled out.
13. Wild factor bound
Verifying the limit of x^2 at 2 with delta at most 1 bounds |x + 2| by a specific integer. What integer? Answer with a number.
14. Continuity's extra freedom
Continuity's definition relaxes one feature of the limit definition by allowing x equal to a. How many strict inequalities of the form 0 less than |x - a| appear in the continuity version? Answer with a number.