The formal definition of Limit — epsilon-delta

A Formal Definition of Limit:
Let f(x) be defined on a open interval about x_{0}, except possibly at x_{0} itself. We say that f(x) approaches the limit L as x approaches x_{0}, and write

\lim_{x \rightarrow x_{0}} f(x)=L,

if for every number \in > 0, there exists a corresponding number \delta >0, such that for x,

0<|x-x_{0}|<\delta \Longrightarrow |f(x)-L|< \in.

Note: In the definition, \delta is a function of \in, or in other words, \delta depends on \in.

This definition is useful to prove important theorems about limits, that is, we can then easily evaluate the limits of complicated functions, whose limits are too cumbersome to evaluate from intuitive first principles.

Think of  the definition as follows: We desire an output tolerance so that we need to find appropriate input given by


Show  that \lim_{x \rightarrow 1}=2


Let x_{0}=1, f(x)=5x-3 and L=2 in the definition of limit. For any given \in > 0, we have to find a number \delta > 0 such that if x \neq 1. and x is within distance \delta of

x_{0}, that is, if

0 < |x-1|< \delta, then f(x) is within distance in of L=2, that is,

|f(x)-2|< \in

We find \delta by working backwards from the \in -inequality.

|(5x-3)-2|= |5x-5|<\in, that is,

5|x-1|< \in, that is,

|x-1|< \in/5.

Important note: The value of \delta=\in/5 is not  the only value that will make

0 < |x-1|< \delta imply |5x-5|<\in. Any smaller positive \delta will do as well. The definition does not ask for a “best” positive \delta, just one that will work.

Any questions, comments are most welcome…

More later…

Nalin Pithwa

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