Difference between revisions of "Limit (mathematics)"
(I like your changes, but an "infinitely small ball" is not intuitive :-)) |
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| − | The concept of the '''limit''' is | + | {{Template:Math-h}} |
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| + | The concept of the '''limit''' is the cornerstone of [[calculus]], [[analysis]], and [[topology]]. At the simplest intuitive level, the limit of a function at a point is the value that the function "approaches" as its argument "approaches" that point. As we will see, that is an unsatisfactory definition, and a much more careful definition is required. | ||
==Limit of a Sequence== | ==Limit of a Sequence== | ||
Revision as of 03:05, August 17, 2008
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
The concept of the limit is the cornerstone of calculus, analysis, and topology. At the simplest intuitive level, the limit of a function at a point is the value that the function "approaches" as its argument "approaches" that point. As we will see, that is an unsatisfactory definition, and a much more careful definition is required.
Limit of a Sequence
Let 
be a sequence of real numbers. We say that this sequence has a limit 
, i.e.,
if for any 
, there exists a number 
, such that
for every 
. Intuitively, this means that if you take an interval as small as you like, centered at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval.
Limit of a Function
Let 
be a real valued function in one variable. We say that
if for any error 
, we can find a sufficiently small neighborhood of 
so that is within of the value 
for any 
in that neighborhood. One says that the limit of f(x) at p exists and is equal to L.
The standard, though more verbose, way of saying this is that for any there exists a sufficiently small 
such that 
whenever 
.
Intuitively, this means that as the variable 
approaches the value , the function tends to the value .
