181
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + − − ==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 22:
Line 15: − + Line 92:
Line 85: − + Line 103:
Line 96: − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − + +
Had been left unfinished.
{{Template:Math-h}}
{{Math-h}}
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. But 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. But that is an unsatisfactory definition, and a much more careful definition is required.
==Limit of a Function==
==Limit of a Function==
Now try an example that isn't trivial. Let
Now try an example that isn't trivial. Let
:<math>f(x) = x^x\,</math>
:<math>f(x) = x^x\,</math>
This function is well-defined for x>0, using the general definition that involves the [[exponential]] and [[natural logarithm]] functions. We can calculate f(x) for various values of x:
This function is well-defined for x>0, using the general definition that involves the [[exponential]] and [[natural logarithm]] functions. But we can calculate f(x) for other values:
f(5) = 3125
f(5) = 3125
f(2) = 4
f(2) = 4
That is,
That is,
:For every M > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) > M\,</math>
:For every M > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, f(x) > M\,</math>
Consider the function
Consider the function
:<math>\lim_{x\to X}f(x) = - \infty\,</math>
:<math>\lim_{x\to X}f(x) = - \infty\,</math>
means
means
:For every M < 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) < M.</math>
:For every M < 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, f(x) < ,</math>
==The limit of f(x) as x Goes to Infinity==
Infinity can also make an appearance in the function's domain, as in "The limit of f(x), as x approaches infinity, is 3." What that means is that we replace the
:<math>0 < |x - X| < \delta\,</math>
with
:<math>x > M\,</math>
That is,
:For every ε > 0, there is an M such that, whenever <math>x > M, |f(x)-Y| < \varepsilon\,</math>
Consider the function
:<math>f(x) = \frac{3x+5}{x}\,</math>
If you graph this, it goes off to a value of 3 toward the right. We have:
:<math>\lim_{x\to \infty}f(x) = 3\,</math>
If we set <math>\varepsilon = 10^{-10}\,</math>, <math>M = 10^{11}\,</math> will win the bet.
We can do a similar thing for the limit as x goes to minus infinity.
==The limit of f(x) as x Goes to Infinity is Infinity==
A function like <math>f(x) = e^x\,</math> combines both inifinites. We write
:<math>\lim_{x\to \infty}e^x = \infty\,</math>
Here we replace both δ and ε:
:For every N > 0, there is an M such that, whenever <math>x > M, f(x) > N\,</math>
==Limit of a Sequence==
An important type of limit is the limit of a [[sequence]] of numbers <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math>. When we say that this sequence has a limit A, we are essentially using the definition above for x going to infinity, but using n instead of x:
:<math>\lim_{n\to \infty}a_n = A,\,</math>
means that
:For every ε > 0, there is an M such that, whenever <math>n > M, |a_n-A| < \varepsilon\,</math>
When a sequence has a limit like that, we say that the sequence '''converges''' to that limit.
==Infinite Sums==
It is important to distinguish between the '''limit of the sequence''', as defined above, and the much more commonly used '''limit of the partial sums of the sequence'''. If we have a sequence <math>(a_n) = (a_1, a_2, a_3, ...)</math>, we also have a sequence of "partial sums", the n<sup>th</sup> item of which is the sum of the first n items in the original sequence:
:<math>s_n = \sum_{i=1}^n a_n\,</math>
If <math>s_n\,</math> converges to A, we say that the '''infinite sum''' <math>a_n\,</math> converges to A, written like this:
:<math>\sum_{i=1}^\infty a_n = A\,</math>
For example, the sequence
:<math>a_n = \frac{1}{n!}\,</math>
(where the exclamation point is the [[factorial]] operation)
converges to the number known as "e":
:<math>\sum_{i=1}^\infty \frac{1}{n!} = e\,</math>
The is the only sense in which a summation going to infinity makes sense.
[[category:mathematics]]
[[Category:Mathematics]]
[[Category:Calculus]]