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The Precise Definition of a Limit
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Formal definition of .
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We say that , provided that
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Given , there exists , such that whenever 0 < .
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We say exists if there exists a real number L such that .
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Remark :
If exists, then given , there exists , such that
whenever 0 < and 0 < .
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Intuitively, if exists, say L, then and can be as close as we want to L by taking both and sufficiently close to a (but not equal to a ). So and can be as close as we want by taking both and sufficiently close to a (but not equal to a ).
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Proof of the remark :
Given , there exists , such that such that whenever 0 < . If 0 < and 0 < , then
< .
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Example Show that .
Idea of the proof :
Given , we want to find , such that such that whenever 0 < .
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This tells us one way to find such a , we should solve the inequality . If , then < . Assume further that , we get
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<
and so
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< .
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This implies that we should take to be the minimum of and .
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Note that both and are positive, in fact (why ?).
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Proof : Given , without loss of generality may assume that , let
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( the minimum of and ).
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If 0 < , then
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< <
Hence,
<
and so
< .
Therefore,
.
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And the proof is completed.
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Note that , if then .
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Alternative proof :
Given , let . If 0 < , then . So we have and
< .
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Formal definition of and .
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We say that , provided that
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Given , there exists , such that whenever 0 < .
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We say that , provided that
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Given , there exists , such that whenever 0 < .
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Example Show that .
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Idea of the proof :
Given , we want to find , such that such that whenever 0 < .
Note that if then 0 < , which implies that 0 < .
Proof : Given , let . If < , then 0 < .
So we have
.
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