Is the Concept of Neighborhoods Essential for Understanding Limits in Calculus?

[daniel_i_l]

Is it correct to think about the expresion:
"the limit of f(x) is b when x->a" as saying that for every x that's very close to a but not a (in the deleted neighborhood of a) there is a f(x) that's very close to b (in the neibourhood of b) - or is that not precise enough?

 

[Galileo]

It's definitely not precise enough for a mathematical definition. What do you mean by 'very close to a'. What is 'very close'?
The idea of $\lim_{x\to a}f(x)=b$ is that you can make f(x) as close as you want to b by choosing x close enough (but not equal to) a. By close I mean that the distance |f(x)-b| can be made as small as we want. How small? Smaller than any given positive number.

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

 

[whozum]

Might want to use 'gets closer' instead of 'is close to'

Thats pretty much the epsilon delta method.

 

[daniel_i_l]

Thanks guys for making that clear.