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Difference between revisions of "Limit"

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The notion of '''limit''' is considered one of the most important ideas in [[Calculus]] and was the one that took several efforts before it was finally formalized.  
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The notion of '''limit''' is an important idea in [[topology]] that applies to [[calculus]], [[analysis]], and other fields of mathematics.  It took several centuries to articulate this idea and to make it rigorous.
  
 
==Definition==
 
==Definition==
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Although Limit can be defined in several settings, we will give the definition for an ordinary ([[Real numbers|real]] to real) function.
 
  
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Let <math>A\subset\mathbb{R}</math>
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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.
  
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Let <math>c</math> be a [[Cluster point|cluster point]] of <math>A</math>
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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:A\rightarrow\mathbb{R}</math>  
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: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>\espilon >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>
  
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Let <math>L\in\mathbb{R}</math>
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However, most theorems on real limits apply to limits in general, with identical proofs.
  
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We say that <math>\lim_{x\rightarrow c}f(x)=L</math> iff
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== Uniqueness ==
  
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<math>\forall\epsilon>0\;\;\;\exists\delta>0</math> such that
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The notation <math>\lim_{x\to c}f(x) = L</math> would only be justifiable if <math>L</math> were unique.  Fortunately, this is the case.
  
−
<math>|x-c|<\delta\implies|f(x)-L|<\epsilon</math>
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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.
 +
 
 +
==Existence of Limits==
 +
 
 +
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>.
  
 
==Intuitive Meaning==
 
==Intuitive Meaning==
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The formal definition of a limit given above is not necessarily easy to understand.  We can instead offer the following informal explanation: a limit is the value to which the function grows close.  For example, <math>\lim_{x\rightarrow 2}x^2=4</math>, 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 is exactly equal to the value of the function.  That is, <math>\lim_{x\rightarrow c} f(x) = f(c)</math>.  This is because the function we chose was a [[continuous function]]. However, not all functions have this property.  For example, consider the function <math>f(x)</math> over the reals defined to be 0 if <math>x\neq 0</math> and 1 if <math>x=0</math>.  Although the value of the function at 0 is 1, the limit <math>\lim_{x\rightarrow 0}f(x)</math> is, in fact, zero.  Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, <math>f(x)</math> will always be close to (in fact equal to) zero.  Note that if our definition required only that <math>|x-c|<\delta</math>, the limit of this function would not exist.
+
 
 +
Many people new to calculus have difficulty understanding the limit's formal definition.  We can instead offer the following informal explanation: a limit is the value to which the function grows close.  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 is equal to the value of the function.  That is, <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 reals 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</math> at 0 is 1, the limit <math>\lim_{x\rightarrow 0}f(x)</math> is, in fact, zero.  Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, <math>f(x)</math> will always be close to (in fact equal to) zero.  Note that if our definition required only that <math>|x-c|<\delta</math>, the limit of this function would not exist.
  
 
==Left and Right Hand Limits==
 
==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>.
 
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>.
 
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>.
−
 
−
==Existence of Limits==
 
−
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>.
 
  
 
A limit exists if the left and right hand side limits exist, and are equal.
 
A limit exists if the left and right hand side limits exist, and are equal.
  
 
==Sequential Criterion==
 
==Sequential Criterion==
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Let <math>A\subset\mathbb{R}</math>, Let <math>c</math> be a [[Cluster point|cluster point]] of <math>A</math>, Let <math>f:A\rightarrow\mathbb{R}</math> and let Let <math>L\in\mathbb{R}</math>
 
  
−
Then
+
Let <math>A\subset\mathbb{R}</math>, Let <math>c</math> be a [[cluster point]] of <math>A</math>, Let <math>f:A\rightarrow\mathbb{R}</math> and let Let <math>L\in\mathbb{R}</math>
  
−
(1)<math>\lim_{x\rightarrow c}f(x)=L</math> if and only if
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Then <math>\lim_{x\rightarrow c}f(x)=L</math> if for every [[sequence]] <math>\left\langle x_n \right\rangle</math> that converges to <math>c</math>, the sequence <math>\left\langle f(x_n) \right\rangle</math> converges to <math>L</math>.
  
−
(2)<math>\forall</math> [[Sequence|sequence]] <math>\left\langle x_n \right\rangle</math> that converges to <math>c</math>, the sequence <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-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>.
−
*If a limit exists, it is unique.
 
  
 
==See also==
 
==See also==
Line 55: Line 82:
 
*[[Neighbourhoods]]
 
*[[Neighbourhoods]]
  
 +
[[Category:Topology]]
 +
[[Category:Analysis]]
 
[[Category:Calculus]]
 
[[Category:Calculus]]
 
[[Category:Definition]]
 
[[Category:Definition]]

Revision as of 17:24, 7 May 2008

The notion of limit is an important idea in topology that applies to calculus, analysis, and other fields of mathematics. It took several centuries to articulate this idea and to make it rigorous.

Definition

Let $A$ and $B$ be metric spaces, let $A'$ be a subspace of $A$, and, let $f$ be a function from $A'$ to $B$. Let $c$ be a limit point of $A'$. (This means that in the metric space $A$, there are elements of $A'$ arbitrarily close to $c$.) Let $L$ be an element of $B$. We say \[\lim_{x\to c} f(x) = L,\] (that is, the limit of $f(x)$ as $x$ goes to $c$ equals $L$) if for every positive real $\epsilon$ there exists a positive real $\delta$ for which \[0 < d_A(x,c) < \delta\] implies \[d_B(f(x),L) < \epsilon\] for all $x \in A'$. Here $d_A$ and $d_B$ are the distance functions of $A$ and $B$, respectively.

In analysis and calculus, usually $A$ and $B$ are both either the set of reals $\mathbb{R}$ or complex numbers $\mathbb{C}$. In this case, the distance functions $d_A(a,b)$ and $d_B(a,b)$ are both simply $|a-b|$. We then obtain the following definition commonly found in calculus textbooks:

Let $f$ 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 $L$,

\[\lim_{x\to c} f(x) =L\]

if for every $\espilon >0$ (Error compiling LaTeX. Unknown error_msg) there exists a $\delta>0$ such that

\[0 < |x-c| < \delta \quad \text{implies} \quad |f(x) - L| < \epsilon .\]

However, most theorems on real limits apply to limits in general, with identical proofs.

Uniqueness

The notation $\lim_{x\to c}f(x) = L$ would only be justifiable if $L$ were unique. Fortunately, this is the case.

Indeed, suppose that $L'$ is also $\lim_{x\to c}f(x)$, and that $L \neq L'$. Since $d_B(L,L') >0$, we can pick a positive real $\epsilon < d_B(L,L')/2$. But for any $y \in L$, \[d_B(L,y) + d_B(L',y) \ge d_B(L,L'),\] so no $y$ can simultaneously satisfy the conditions \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*} a contradiction. Therefore limits are unique, as we wanted.

Existence of Limits

Limits do not always exist. For example $\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since, in fact, there exists no $\epsilon$ for which there exists $\delta$ satisfying the definition's conditions, since $\left|\frac{1}{x}\right|$ grows arbitrarily large as $x$ approaches 0. However, it is possible for $\lim_{x\rightarrow c} f(x)$ not to exist even when $f$ is defined at $c$. For example, consider the Dirichlet function, $D(x)$, defined to be 0 when $x$ is irrational, and 1 when $x$ is rational. Here, $\lim_{x\rightarrow c}D(x)$ does not exist for any value of $c$. Alternatively, limits can exist where a function is not defined, as for the function $f(x)$ defined to be 1, but only for nonzero reals. Here, $\lim_{x\rightarrow 0}f(x)=1$, since for $x$ arbitrarily close to 0, $f(x)=1$.

Intuitive Meaning

Many people new to calculus have difficulty understanding the limit's formal definition. We can instead offer the following informal explanation: a limit is the value to which the function grows close. For example, \[\lim_{x\to 2}x^2=4,\] because whenever $x$ is close to 2, the function $f(x)=x^2$ grows close to 4.

In this case, the limit of the function is equal to the value of the function. That is, $\lim_{x\rightarrow c} f(x) = f(c)$. This is because the function we chose was continuous at $c$.

However, not all functions have this property. For example, consider the function $f(x)$ over the reals defined as follows: \[f(x) = \begin{cases} 0 & \text{if } x\neq 0 , \\ 1 & \text{if } x=0 . \end{cases}\] Although the value of the function $f$ at 0 is 1, the limit $\lim_{x\rightarrow 0}f(x)$ is, in fact, zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, $f(x)$ will always be close to (in fact equal to) zero. Note that if our definition required only that $|x-c|<\delta$, the limit of this function would not exist.

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 $\lim_{x\to c^{-}} f(x)$, and the right hand limit is denoted as $\lim_{x\to c^{+}} f(x)$.

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) $f(x) = \lfloor x \rfloor$, we have $\lim_{x\to 0^{+}} \lfloor x \rfloor = 0$, while $\lim_{x\to 0^{-}} \lfloor x \rfloor = -1$.

A limit exists if the left and right hand side limits exist, and are equal.

Sequential Criterion

Let $A\subset\mathbb{R}$, Let $c$ be a cluster point of $A$, Let $f:A\rightarrow\mathbb{R}$ and let Let $L\in\mathbb{R}$

Then $\lim_{x\rightarrow c}f(x)=L$ if for every sequence $\left\langle x_n \right\rangle$ that converges to $c$, the sequence $\left\langle f(x_n) \right\rangle$ converges to $L$.

Other Properties

Let $f$ and $g$ be real functions. Then:

  • $\lim(f+g)(x)=\lim f(x)+\lim g(x)$
  • $\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
  • $\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$ given that $\lim g(x)\ne 0$.

See also

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