Limit Definition,

[Bitupon]

I have a doubt regarding the definition of limit found in most calculus

books.

 

The definition of limit says that f(x) approaches the limit L as x

approaches c if, for every number e>0 there exists a corresponding

number d>0 such that

for all x 0<│x-c│< d => │f(x)-L│< e.

 

Can’t we replace │f(x)-L│< e with 0< │f(x)-L│< e? If not then why?

[Erasmus00]

You don't need to, its redundant. The absolute value of anything is always greater than 0.

-Will

[Qfwfq]

I think, Erasmus, he means instead of specifying e>0.

 

The difference Bitupon is that logically, if you say "for every number" without specifying strictly greater than zero, the next bit would have to hold even for zero and negative epsilon values.

 

A more essential alternative is to use the definition of "open neighborhood" in topology but the disequalities give the same effect. High school calculus books use them because it is simpler, although topology gives a better overall view, once one has climbed the hill. :hihi: