Sounds like a very good teacher!

I once taught a course called "Quantitative Methods for Economics and Business Administration" using a text book assigned by the Business Administration department. On one page of this book we were given three laws of limits:

1) If and then

2) If and then

3) if andandM is not 0, then

On the very next page, they introduced the derivative as the limit of the difference quotient: , completely ignoring the fact that the denominator necessarily goes to 0 so this limit could NEVER be done using only those laws!

Yes, that property isveryimportant, and fairly easy to prove.

Let . Then, by definition of limit, given any there exist such that if then . Note the "<" symbol at the left of |x-a|. It does not matter what happens at x= a because then |x-a|= 0! (I am using the subscript on to distinguish if from the given in the problem.)

Now, given any , we can use thesmallerof and to prove that . If 0< |x- a|< the smaller of and , thenbothand f(x)= g(x) are true. So we can replace f(x) by g(x) in that expression: and we are done!