Definition of Limit
Main Concept

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 d > 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 d so that .)

Example 1


Prove:





Note: .




Remember you are trying to prove that:
For all , there exists a such that:
if then .

Step1: Determine what to choose for



Substitute all values into .









The relation has been simplified to the form , if you choose .




Step 2: Assume , and use that relation to prove that .



Substitute values for and .






















Follow the instructions, using different functions , values of , e and d to observe graphically why the proof works.

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