Limit
For a real function and some value
,
(said, "the limit of
at
as
goes to
) equals
iff for every
there exists a
such that if
, then
.
Intuitive Meaning
The definition of a limit is a difficult thing to grasp, so many books give an intuitive definition first: a limit is the value to which the rest of the function grows closer. For example, , because as
grows arbitrarily close to 2 from either direction, the function
grows arbitrarily close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is,
. This is because the function we chose was a continuous function. Unfortunately, this does not hold true in general. For example, consider the function
over the reals defined to be 0 if
and 1 if
. Although the value of the function at 0 is 1, the limit
is, in fact, zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero,
will always be close to (in fact equal to) zero. Note that if our definition required only that
, the limit of this function would not exist.
Existence of Limits
Limits do not always exist. For example does not exist, since, in fact, there exists no
for which there exists
satisfying the definition's conditions, since
grows arbitrarily large as
approaches 0. However, it is possible for
not to exist even when
is defined at
. For example, consider the Dirichlet function,
, defined to be 0 when
is irrational, and 1 when
is rational. Here,
does not exist for any value of
. Alternatively, limits can exist where a function is not defined, as for the function
defined to be 1, but only for nonzero reals. Here,
, since for
arbitrarily close to 0,
.
Other Properties
Let and
be real functions. Then:
- If a limit exists, it is unique.