Difference between revisions of "Limit"

Revision as of 09:27, 20 April 2008

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.

Definition

Although Limit can be defined in several settings, we will give the definition for an ordinary (real to real) function.

Let $A\subset\mathbb{R}$

Let $c$ be a cluster point of $A$

Let $f:A\rightarrow\mathbb{R}$

Let $L\in\mathbb{R}$

We say that $\lim_{x\rightarrow c}f(x)=L$ iff

$\forall\epsilon>0\;\;\;\exists\delta>0$ such that

$|x-c|<\delta\implies|f(x)-L|<\epsilon$

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 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 $\lim_{x\to 0^{-}} \lfloor x \rfloor = -1$ .

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.

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

(1) $\lim_{x\rightarrow c}f(x)=L$ if and only if

(2) $\forall$ 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:

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

@@ Line 48: / Line 48: @@
 *If a limit exists, it is unique.
-==See Also==
+==See also==
 *[[Calculus]]
 *[[Derivative]]

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