I thought I understood how to show that a function has a limit using the limit definition, but my book is doing a problem in a way I can't follow. The question is "prove that the limit of x^2 as x approaches 2 is 4"
The book starts with the epsilon equation
and they show that
which I can understand (though somehow they assume epsilon is less than 4)
Then they say that they want to find delta. They show that delta is
Which I am now able to follow although it took awhile to see why this is true. Up to this point I can understand what is going on. This is all the "scratch work" if I am correct. Now they have to actually prove the limit and they say the following...
"If delta has this or any smaller positive value, the inequality
will place x between
This completes the proof."
I don't understand how this completes the proof. I can prove the limit a different way but it is disheartening that I can't follow this. Can anyone explain how their work proves the limit. Thanks.