Difference between revisions of "Limit"

Revision as of 13:56, 8 January 2008

Given a real or complex function $f(x)$ and some value $c$ , the limit $\lim_{x\to c} f(x)$ of $f(x)$ as $x$ goes to $c$ is defined to be equal to the (real or complex) number $L$ if and only if for every $\epsilon > 0$ there exists a $\delta \in \mathbb R$ such that if $0<|x-c|<\delta$ then $|f(x)-L|< \epsilon$ .

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, $\lim_{x\rightarrow 2}x^2=4$ , 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 exactly 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 a continuous function. However, not all functions have this property. For example, consider the function $f(x)$ over the reals defined to be 0 if $x\neq 0$ and 1 if $x=0$ . Although the value of the function 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

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 $\lim_{x\to c^{-}} f(x)$ , and the right hand limit is in the form $\lim_{x\to c^{+}} f(x)$ , since the left side of the Cartesian Coordinate plane is negative, and the right hand side is positive.

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, $f(x)=1$ .

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

Other Properties

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

$\lim(f+g)(x)=\lim f(x)+\lim g(x)$
$\lim(f-g)(x)=\lim f(x)-\lim g(x)$
$\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
$\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$ given that $\lim g(x)\ne 0$ .
If a limit exists, it is unique.

@@ Line 3: / Line 3: @@
 ==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.
+==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]] [[plane]] is negative, and the right hand side is positive.
 ==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.
 ==Other Properties==

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