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Limit
Let 
and 
be metric spaces
and let 
be a limit point
of 
. Suppose that
is a function
defined everywhere except at
. For 
, we say the limit of 
as 
approaches 
is equal to 
, or
, there exists a real
number
such that, whenever 
with
, then
.
The formal definition of limit as given above has a well-deserved reputation for being notoriously hard for inexperienced students to master. There is no easy fix for this problem, since the concept of a limit is inherently difficult to state precisely (and indeed wasn't even accomplished historically until the 1800's by Cauchy, well after the development of calculus in the 1600's by Newton and Leibniz). However, there are number of related definitions, which, taken together, may shed some light on the nature of the concept.
- The notion of a limit can be generalized to mappings
between arbitrary
topological spaces, under some mild restrictions. In this context we say that
if 
is a limit point of 
and, for every
neighborhood
of 
(in 
), there is a deleted neighborhood
of 
(in 
) which is mapped into 
by 
. One also requires that the range
be Hausdorff
(or at least 
) in order
to ensure that limits, when they exist, are unique.
- Let
be a sequence
of elements in a metric
space 
. We say that 
is the limit of the sequence, if
for every
there exists a natural number
such
that
for all natural numbers 
.
- The definition of the limit of a mapping can be based on the
limit of a sequence. To wit,
if and only
if, for every sequence of points
in 
converging to 
(that is, 
,
), the sequence of points
in 
converges
to 
.
and 
are frequently taken to be Euclidean spaces
and
, in which case the distance
functions 
and 
cited above are just Euclidean distance.