Difference between revisions of "Limit"

Latest revision as of 17:39, 28 September 2024

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

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, $\lim_{x\to 2}x^2=4$ 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 happens to equal the value of the function ( $\lim_{x\rightarrow c} f(x) = f(c)$ ). 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 $f(x)$ at $x = 0$ is $1$ , the limit $\lim_{x\rightarrow 0} f(x)$ 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 $x$ is very close (but not equal!) to zero, $f(x)$ will always be close to (in fact equal to) zero.

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 terms of our informal definition, $\epsilon$ is a measure of "how close" we want $f(x)$ to be to its limit value. Then the formal definition says that no matter how close we want to be (for any $\epsilon > 0$ ), we can make our variable close enough (within a distance $\delta$ , for some $\delta$ ) to $c$ to achieve our goal.

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 $\epsilon >0$ 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.

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, $f(x)=1$ .

Uniqueness

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

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.

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 $A\subset\mathbb{R}$ and let $c$ be a cluster point of $A$ . A function $f : A \rightarrow \mathbb{R}$ has a limit $L = \lim_{x \rightarrow c} f(x)$ if for every sequence $\left\langle x_n \right\rangle$ that converges to $c$ , $\left\langle f(x_n) \right\rangle$ 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 $\lim g(x)\ne 0$ .

@@ Line 1: / Line 1: @@
-Given a [[Real numbers|real]] or [[complex numbers | complex]] [[function]] <math>f(x)</math> and some value <math>c</math>, the '''limit''' <math>\lim_{x\to c} f(x)</math> of <math>f(x)</math> as <math>x</math> goes to <math>c</math> is defined to be equal to the (real or complex) number <math>L</math> if and only if for every <math>\epsilon > 0</math> there exists a <math>\delta \in \mathbb R</math> such that if <math>0<|x-c|<\delta</math> then <math>|f(x)-L|< \epsilon</math>.
+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==
-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#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==
+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==
-Left and Right hand limits are what they seem to be: limits from the left, and limits from the right. The left hand limit is in the form <math>\lim_{x\to c^{-}} f(x)</math>, and the right hand limit is in the form <math>\lim_{x\to c^{+}} f(x)</math>, since the left side of the [[Cartesian coordinate system]] is negative, and the right hand side is positive.
-==Existence of Limits==
+In this section, we consider limits of functions whose domain and range are both subsets of the set of reals.
-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>.
+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==
 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\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==
 *[[Calculus]]
 *[[Derivative]]
 *[[L'Hopital's Rule]]
 *[[Squeeze Play Theorem]]
+*[[Neighborhoods]]
+[[Category:Topology]]
+[[Category:Analysis]]
 [[Category:Calculus]]
 [[Category:Definition]]
+[[category:Mathematics]]

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