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Line 2: − − ==Limit of a Sequence== − Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a</math>, i.e., − <math> − \lim_{n\to \infty} = a, − </math> − if for any <math>\varepsilon > 0</math>, there exists a number <math>N</math>, such that − <math> − |a_n - a| < \varepsilon</math> for every <math>n > N</math>. Intuitively, this means that if you take an [[interval]] as small as you like, [[center]]ed at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval. Line 70:
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Function should be first. It is more fundamental, even if it might be considered harder.
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.
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 Function==
==Limit of a Function==
We still haven't proved that the limit is actually 1. We just gave some accurate calculations strongly suggesting that it is. In fact it is, and the proof requires a few theorems about limits, [continuous function]]s, and the [[exponential]] and [[natural logarithm]] functions.
We still haven't proved that the limit is actually 1. We just gave some accurate calculations strongly suggesting that it is. In fact it is, and the proof requires a few theorems about limits, [continuous function]]s, and the [[exponential]] and [[natural logarithm]] functions.
==Limit of a Sequence==
Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a</math>, i.e.,
<math>
\lim_{n\to \infty} = a,
</math>
if for any <math>\varepsilon > 0</math>, there exists a number <math>N</math>, such that
<math>
|a_n - a| < \varepsilon</math> for every <math>n > N</math>. Intuitively, this means that if you take an [[interval]] as small as you like, [[center]]ed at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval.
[[category:mathematics]]
[[category:mathematics]]