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Math 1304 Calculus I 2.4 – Definition for Limits
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Recall Notation for Limits The reads: The limit of f(x), as x approaches a, is equal to L Meaning: As x gets closer to a, f(x) gets closer to L.
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Questions Can we make it more precise? Can we use this more precise definition to prove the rules?
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Distance What do we mean by “As x gets closer to a, f(x) gets closer to L”. Can we measure how close? (We use distance.) What’s the distance? |x-a| is distance between x and a |f(x)-L| is distance between f(x) and L a x f(x)L
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How close? Measuring distance: argument and value. a x f(x)L Distance of arguments =|x - a| Distance of values = |f(x)–L| Can we make the distance between the values of f and L small by making the distance between x and a small? Turn this into a bargain: Given any >0, find >0 such that |f(x)-L|< , whenever 0<|x-a|< .
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Formal Definition of Limits Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is L, if for each positive real number >0 there is a positive real >0 such that |f(x)-L|< , whenever 0<|x-a|< . When this happens we write
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Other ways to say it Given >0 there is >0 such that |f(x)-L|< , whenever 0<|x-a|< . Given >0 there is >0 such that 0<|x-a|< |f(x)-L|<
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Definition in terms of intervals Given >0 there is >0 such that a- < x < a+ and x≠a L- < f(x) < L+ a a+ a- L L+ L -
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Picture a a+ a- L L+ L- x f(x)
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Example Using the above definition, prove that f(x) = 2x+1 has limit 5 at x=2 Method: guess and then verify. Work backwards: compute and estimate the distance |f(x)-L| in terms of the distance |x-a| Use the estimate to guess a delta, given the epsilon.
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Nearby Behavior Note that if two functions agree except at a point a, they have the same limit at a, if it exists. Stronger result: If two functions agree on an open interval around a point a, but not necessarily at a, and one has a limit at a, then they have the same limit at a.
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Proof of Rules Can prove the above rules from this definition. Example: the sum rule (in class) Note: we need the triangle inequality: |A+B| |A|+|B|
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One-sided limits: Left Definition of left-hand limit if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a- <x<a.
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One-sided limits: Right Definition of left-hand limit if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a<x<a+ .
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Limits of plus infinity Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is , if for each real number M there is a positive real >0 such that f(x)>M, whenever 0<|x-a|< . When this happens we write
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Limits of minus infinity Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is - , if for each real number M there is a positive real >0 such that f(x)<M, whenever 0<|x-a|< . When this happens we write
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