Cauchy and Heine Definitions of Limit
Let f(x) be a function that is defined on an open interval X containing x=a. (The value f(a) need not be defined.)
The number L is called the limit of function f(x) as xâ†’a if and only if, for every Îµ>0 there exists Î´>0 such that
f(x)âˆ’L<Îµ,
whenever
0<xâˆ’a<Î´.
This definition is known as Îµâˆ’Î´âˆ’ or Cauchy definition for limit.
Thereâ€™s also the Heine definition of the limit of a function, which states that a function f(x) has a limit L at x=a, if for every sequence {xn}, which has a limit at a, the sequence f(xn) has a limit L. The Heine and Cauchy definitions of limit of a function are equivalent.
In formulas, a limit of a function is usually written as
and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (â†’), as in
