# What is the definition of limit in calculus?

##### 1 Answer

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:

**Definition**

Let

Then the limit as

if and only if

for every

if

**That is the end of the definition**

**Comments**

Tlhe following version is a bit more "wordy", but it is clearer to many.

for every

there is a

for which the following is true:

if

then

**An acceptable rephrasing** of that "if . . ., tlhen . . . " is:

If

then

**The Game**

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 **chose** to

(For nice functions this last bit about