The Precise Definition of Limit

1. The Precise Definition of Limit Augustin-Louis Cauchy Karl Weierstrass

2. Augustin-Louis Cauchy 1789-1857 Karl Weierstrass 1815-1897

3. Vague “Intuitive”Definition From the textbook: Def. We write if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a.

4. Hybrid Definition Given in class: Def. We write if, for any ε>0, we can make f(x) stay within a distance of ε from L by requiring x to be close enough to a but not equal to a. We will now progress step by step to the precise definition. All statements that follow are definitions of the meaning of .

5. For any ε>0, we can make f(x) stay within a distance of ε from L by requiring x to be close enough to a but not equal to a. Q. What do we mean by “close enough”? A. Within some fixed distance from a Let’s call a sufficient bound on the distance δ. δ needs to be positive as well. For any ε>0, there is a corresponding δ>0 such thatwe can make f(x) stay within a distance of ε from L by requiring x to be within a distance of δ from a but not equal to a.

6. For any ε>0, there is a corresponding δ>0 such that we can make f(x) stay within a distance of ε from L by requiring x to be within a distance of δ from a but not equal to a. Let’s rephrase this more directly in terms of the distances between the numbers. In particular, the requirement that x is not equal to a can be rephrased by saying the distance between them is not 0 (i.e. greater than 0). For any ε>0, there is a corresponding δ>0 such that the distance between f(x) and L is less than εwhenever the distance between x and a is less than δ from a but greater than 0.

7. For any ε>0, there is a corresponding δ>0 such that the distance between f(x) and L is less than εwhenever the distance between x and a is less than δ from a but greater than 0. How do we determine the distance between two real numbers? Subtract one from the other. We also include absolute values on each difference since distances should be nonnegative. For any ε>0, there is a corresponding δ>0 such that |f(x) – L| <εfor all x such that |x – a|< δ and |x – a|> 0.

8. For any ε>0, there is a corresponding δ>0 such that |f(x) – L| < ε for all x such that |x – a|< δ and |x – a|> 0. Finally, we combine the last two inequalities. For any ε>0, there is a corresponding δ>0 such that |f(x) – L| < ε for all x such that 0<|x – a|< δ. This is the precise definition of a limit!

9. The Precise Definition of Limit The expression means that for any ε>0, there is a corresponding δ>0 such that |f(x) – L| < ε for all x such that 0<|x – a|< δ. This is also called the ε- δ definition of limit.

10. Picture ε δ (on each side) Copied from www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

11. Think dynamically! • As ε gets smaller, δ must get smaller too (usually). • How does δ depend on ε? http://www2.scc-fl.edu/lvosbury/CalculusI_Folder/EpsilonDelta.htm