What is the definition of limit in calculus?
There are several ways of stating the definition of the limit of a function. In order for an alternative to be acceptable it must give the same results as the other accepted definitions. Those other definitions are accepted exactly because they do give the same results.
The definition of the limit of a function given in textbooks used for Calculus I in the U.S. is some version of:
Then the limit as
if and only if
That is the end of the definition
Tlhe following version is a bit more "wordy", but it is clearer to many.
there is a
for which the following is true:
An acceptable rephrasing of that "if . . ., tlhen . . . " is:
I claim that
What I am claiming is that:
if someone else chooses how close I need to make
then I'll show that if you start with an
Because I am only making a claim about values of
(For nice functions this last bit about