Definition of Limit
Main Concept
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The precise definition of a limit states that:
Let be a function defined on an open interval containing (except possibly at ) and let be a real number.
Define the limit of at to be , or write
if the following statement is true:
For any e > 0 there is a δ > 0 such that whenever
then also
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Suppose you want to prove that a certain function has a limit. What exactly needs to be determined?
An input range in which there is a corresponding output. (A positive δ so that .)
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Example 1
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Prove:
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Note: .
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Remember you are trying to prove that:
For all , there exists a such that:
if then .
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Step1: Determine what to choose for
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Substitute all values into .
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The relation has been simplified to the form , if you choose .
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Step 2: Assume , and use that relation to prove that .
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Substitute values for and .
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Follow the instructions, using different functions , values of , e and δ to observe graphically why the proof works.
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