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Definition of Limit
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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
 .
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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