Hey everyone. I am having trouble understanding the following statements from my textbook (Essential Calculus by Stelwart)

"..We showed that iffis differentiable ata, then

where as "

What does the epsilon delta x have to do with this equation? I divided everything by delta x and got this:

I'm assuming lim is involved so this is what it would be:

So is epsilon error? If so, why would there be error? I'm definitely confused.

The book goes on to say more:

Now consider a function of two variables, z = f(x,y), and the suppose x changes from a to and y changes from b to . Then the correspondingincrementof z is

Thus, the increment delta z represents the change in the value of f when (x,y) changes from (a,b) to . by analogy with (5) (which is the first equation with the epsilon crap) we define the differentiability of a function of two variables as follows:

If z = f(x,y), then f is differentiable at (a,b) if delta x can be expressed in the form

where and as .

It makes sense except the epsilon stuff.

Any help is appreciated and thanks in advance!