Difference between revisions of "Limit"
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− | + | The '''limit''' is the key concept that separates [[calculus]] from elementary mathematics such as [[arithmetic]], [[elementary algebra]] or [[Euclidean geometry]]. It also arises and plays an important role in the more general settings of [[topology]], [[analysis]], and other fields of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous. | |
==Intuitive Meaning== | ==Intuitive Meaning== | ||
− | + | ||
+ | Many people new to calculus have difficulty understanding the [[Limit#Definition | formal definition of a limit]]. Thus we begin with an informal explanation: a limit is the value to which a function grows close when its argument is near (but not at!) a particular value. For example, | ||
+ | <cmath>\lim_{x\to 2}x^2=4</cmath> because whenever <math>x</math> is close to 2, the function <math>f(x)=x^2</math> grows close to 4. | ||
+ | |||
+ | In this case, the limit of the function happens to equal the value of the function (<math>\lim_{x\rightarrow c} f(x) = f(c)</math>). This is because the function we chose was [[continuous]] at <math>c</math>. | ||
+ | |||
+ | However, not all functions have this property. For example, consider the function <math>f(x)</math> over the [[real]]s defined as follows: <cmath>f(x) = \begin{cases} 0 & \text{if } x\neq 0,\\ 1 & \text{if } x=0. \end{cases}</cmath> Although the value of the function <math>f(x)</math> at <math>x = 0</math> is <math>1</math>, the limit <math>\lim_{x\rightarrow 0} f(x)</math> is, in fact, zero. Intuitively, this is because the limit describes the behavior of the function near (but not at!) the value in question: when <math>x</math> is very close (but not equal!) to zero, <math>f(x)</math> will always be close to (in fact equal to) zero. | ||
+ | |||
+ | ==Definition== | ||
+ | |||
+ | Let <math>A</math> and <math>B</math> be [[metric space]]s, let <math>A'</math> be a [[subspace]] of <math>A</math>, and, let <math>f</math> be a function from <math>A'</math> to <math>B</math>. Let <math>c</math> be a [[limit point]] of <math>A'</math>. (This means that in the metric space <math>A</math>, there are elements of <math>A'</math> arbitrarily close to <math>c</math>.) Let <math>L</math> be an element of <math>B</math>. We say <cmath>\lim_{x\to c} f(x) = L,</cmath> (that is, the limit of <math>f(x)</math> as <math>x</math> goes to <math>c</math> equals <math>L</math>) if for every positive real <math>\epsilon</math> there exists a positive real <math>\delta</math> for which <cmath>0 < d_A(x,c) < \delta</cmath> implies <cmath>d_B(f(x),L) < \epsilon</cmath> for all <math>x \in A'</math>. Here <math>d_A</math> and <math>d_B</math> are the [[distance function]]s of <math>A</math> and <math>B</math>, respectively. | ||
+ | |||
+ | In terms of our informal definition, <math>\epsilon</math> is a measure of "how close" we want <math>f(x)</math> to be to its limit value. Then the formal definition says that no matter how close we want to be (for ''any'' <math>\epsilon > 0</math>), we can make our variable close enough (within a distance <math>\delta</math>, for some <math>\delta</math>) to <math>c</math> to achieve our goal. | ||
+ | |||
+ | In analysis and calculus, usually <math>A</math> and <math>B</math> are both either the set of [[real number | real]]s <math>\mathbb{R}</math> or [[complex number]]s <math>\mathbb{C}</math>. In this case, the distance functions <math>d_A(a,b)</math> and <math>d_B(a,b)</math> are both simply <math>|a-b|</math>. We then obtain the following definition commonly found in calculus textbooks: | ||
+ | |||
+ | :Let <math>f</math> be a function whose [[domain]] is a sub-interval of the real numbers and whose [[codomain]] is the set of reals. For a real number <math>L</math>, | ||
+ | <cmath> \lim_{x\to c} f(x) =L </cmath> | ||
+ | :if for every <math>\epsilon >0</math> there exists a <math>\delta>0</math> such that | ||
+ | <cmath> 0 < |x-c| < \delta \quad \text{implies} \quad |f(x) - L| < \epsilon . </cmath> | ||
+ | |||
+ | However, most theorems on real limits apply to limits in general, with identical proofs. | ||
==Existence of Limits== | ==Existence of Limits== | ||
− | Limits do not always exist. | + | |
+ | Limits do not always exist. For example <math>\lim_{x\rightarrow 0}\frac{1}{x}</math> does not exist, since, in fact, there exists no <math>\epsilon</math> for which there exists <math>\delta</math> satisfying the definition's conditions, since <math>\left|\frac{1}{x}\right|</math> grows arbitrarily large as <math>x</math> approaches 0. However, it is possible for <math>\lim_{x\rightarrow c} f(x)</math> not to exist even when <math>f</math> is defined at <math>c</math>. For example, consider the Dirichlet function, <math>D(x)</math>, defined to be 0 when <math>x</math> is irrational, and 1 when <math>x</math> is rational. Here, <math>\lim_{x\rightarrow c}D(x)</math> does not exist for any value of <math>c</math>. Alternatively, limits can exist where a function is not defined, as for the function <math>f(x)</math> defined to be 1, but only for nonzero reals. Here, <math>\lim_{x\rightarrow 0}f(x)=1</math>, since for <math>x</math> arbitrarily close to 0, <math>f(x)=1</math>. | ||
+ | |||
+ | == Uniqueness == | ||
+ | |||
+ | The notation <math>\lim_{x\to c}f(x) = L</math> would only be justifiable if the limit <math>L</math> were unique. Fortunately, it is always the case that if a limit exists, it is unique. | ||
+ | |||
+ | Indeed, suppose that <math>L'</math> is also <math>\lim_{x\to c}f(x)</math>, and that <math>L \neq L'</math>. Since <math>d_B(L,L') >0</math>, we can pick a positive real <math>\epsilon < d_B(L,L')/2</math>. But for any <math>y \in L</math>, | ||
+ | <cmath> d_B(L,y) + d_B(L',y) \ge d_B(L,L'), </cmath> | ||
+ | so no <math>y</math> can simultaneously satisfy the conditions | ||
+ | <cmath> \begin{align*} | ||
+ | d_B(L,y) &< \epsilon < \frac{d_B(L,L')}{2} \\ | ||
+ | d_B(L',y) &< \epsilon < \frac{d_B(L,L')}{2} , | ||
+ | \end{align*} </cmath> | ||
+ | a contradiction. Therefore limits are unique, as we wanted. | ||
+ | |||
+ | ==Left and Right Hand Limits== | ||
+ | |||
+ | In this section, we consider limits of functions whose domain and range are both subsets of the set of reals. | ||
+ | |||
+ | Left and right hand limits are the limits taken as a point is approached from the left and from the right, respectively. The left hand limit is denoted as <math>\lim_{x\to c^{-}} f(x)</math>, and the right hand limit is denoted as <math>\lim_{x\to c^{+}} f(x)</math>. | ||
+ | |||
+ | If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider the step function (the greatest integer function) <math>f(x) = \lfloor x \rfloor</math>, we have <math>\lim_{x\to 0^{+}} \lfloor x \rfloor = 0</math>, while <math>\lim_{x\to 0^{-}} \lfloor x \rfloor = -1</math>. | ||
+ | |||
+ | A limit exists if the left and right hand side limits exist, and are equal. | ||
+ | |||
+ | ==Sequential Criterion== | ||
+ | |||
+ | Let <math>A\subset\mathbb{R}</math> and let <math>c</math> be a [[cluster point]] of <math>A</math>. A function <math>f : A \rightarrow \mathbb{R}</math> has a limit <math>L = \lim_{x \rightarrow c} f(x)</math> if for every [[sequence]] <math>\left\langle x_n \right\rangle</math> that converges to <math>c</math>, <math>\left\langle f(x_n) \right\rangle</math> converges to <math>L</math>. | ||
==Other Properties== | ==Other Properties== | ||
+ | |||
Let <math>f</math> and <math>g</math> be real functions. Then: | Let <math>f</math> and <math>g</math> be real functions. Then: | ||
*<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> given that <math>\lim g(x)\ne 0</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>. | ||
− | |||
− | ==See | + | ==See also== |
*[[Calculus]] | *[[Calculus]] | ||
*[[Derivative]] | *[[Derivative]] | ||
*[[L'Hopital's Rule]] | *[[L'Hopital's Rule]] | ||
*[[Squeeze Play Theorem]] | *[[Squeeze Play Theorem]] | ||
+ | *[[Neighborhoods]] | ||
− | + | [[Category:Topology]] | |
+ | [[Category:Analysis]] | ||
[[Category:Calculus]] | [[Category:Calculus]] | ||
[[Category:Definition]] | [[Category:Definition]] |
Latest revision as of 13:51, 1 June 2015
The limit is the key concept that separates calculus from elementary mathematics such as arithmetic, elementary algebra or Euclidean geometry. It also arises and plays an important role in the more general settings of topology, analysis, and other fields of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous.
Contents
Intuitive Meaning
Many people new to calculus have difficulty understanding the formal definition of a limit. Thus we begin with an informal explanation: a limit is the value to which a function grows close when its argument is near (but not at!) a particular value. For example,
because whenever
is close to 2, the function
grows close to 4.
In this case, the limit of the function happens to equal the value of the function (). This is because the function we chose was continuous at
.
However, not all functions have this property. For example, consider the function over the reals defined as follows:
Although the value of the function
at
is
, the limit
is, in fact, zero. Intuitively, this is because the limit describes the behavior of the function near (but not at!) the value in question: when
is very close (but not equal!) to zero,
will always be close to (in fact equal to) zero.
Definition
Let and
be metric spaces, let
be a subspace of
, and, let
be a function from
to
. Let
be a limit point of
. (This means that in the metric space
, there are elements of
arbitrarily close to
.) Let
be an element of
. We say
(that is, the limit of
as
goes to
equals
) if for every positive real
there exists a positive real
for which
implies
for all
. Here
and
are the distance functions of
and
, respectively.
In terms of our informal definition, is a measure of "how close" we want
to be to its limit value. Then the formal definition says that no matter how close we want to be (for any
), we can make our variable close enough (within a distance
, for some
) to
to achieve our goal.
In analysis and calculus, usually and
are both either the set of reals
or complex numbers
. In this case, the distance functions
and
are both simply
. We then obtain the following definition commonly found in calculus textbooks:
- Let
be a function whose domain is a sub-interval of the real numbers and whose codomain is the set of reals. For a real number
,
- if for every
there exists a
such that
However, most theorems on real limits apply to limits in general, with identical proofs.
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,
.
Uniqueness
The notation would only be justifiable if the limit
were unique. Fortunately, it is always the case that if a limit exists, it is unique.
Indeed, suppose that is also
, and that
. Since
, we can pick a positive real
. But for any
,
so no
can simultaneously satisfy the conditions
a contradiction. Therefore limits are unique, as we wanted.
Left and Right Hand Limits
In this section, we consider limits of functions whose domain and range are both subsets of the set of reals.
Left and right hand limits are the limits taken as a point is approached from the left and from the right, respectively. The left hand limit is denoted as , and the right hand limit is denoted as
.
If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider the step function (the greatest integer function) , we have
, while
.
A limit exists if the left and right hand side limits exist, and are equal.
Sequential Criterion
Let and let
be a cluster point of
. A function
has a limit
if for every sequence
that converges to
,
converges to
.
Other Properties
Let and
be real functions. Then:
given that
.