Difference between revisions of "Limit"
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*<math>\lim(f-g)(x)=\lim f(x)-\lim g(x)</math> | *<math>\lim(f-g)(x)=\lim f(x)-\lim g(x)</math> | ||
*<math>\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)</math> | *<math>\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)</math> | ||
| − | *<math>\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}</math> | + | *<math>\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}</math> given that <math>\lim g(x)\ne 0</math>. |
* If a limit exists, it is unique. | * If a limit exists, it is unique. | ||
Revision as of 16:05, 6 January 2008
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:
given that 
.- If a limit exists, it is unique.
given that 
.
