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The Precise Definition of Limit Augustin-Louis Cauchy Karl Weierstrass

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Augustin-Louis Cauchy Karl Weierstrass

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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.

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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.

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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 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.

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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 ε whenever the distance between x and a is less than δ from a but greater than 0.

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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| < ε for all x such that |x – a| 0.

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For any ε>0, there is a corresponding δ>0 such that |f(x) – L| 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!

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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.

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Picture Copied from ε δ (on each side)

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Think dynamically! As ε gets smaller, δ must get smaller too (usually). How does δ depend on ε?

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