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'iterated limit in $\mathbb{R}^2$'
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| Title of object: |
iterated limit in $\mathbb{R}^2$ |
| Canonical Name: |
IteratedLimitInMathbbR2 |
| Type: |
Definition |
| Created on: |
2007-08-13 10:45:53 |
| Modified on: |
2007-08-13 20:44:19 |
| Classification: |
msc:26A06, msc:26B12 |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
Let $f$ be a function from a subset $S$ of\, $\mathbb{R}^2$\, to\, $\mathbb{R}$ and\, $(a,\, b)$\, an accumulation point of $S$. The limits
$$\lim_{x\to a}\left(\lim_{y\to b}f(x,\,y)\right) \quad\mbox{and}\quad \lim_{y\to b}\left(\lim_{x\to a}f(x,\,y)\right)$$
are called {\em iterated limits}.\\
\textbf{Example 1.} If\; $\displaystyle f(x,\,y) := \frac{x\sin\frac{1}{x}+y}{x+y}$,\, then
\begin{itemize}
\item $\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right) = \lim_{x\to0}\sin\frac{1}{x}$ does not exist
\item $\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right) = \lim_{y\to0}1 = 1$
\item the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ does not exist.
\end{itemize}
\textbf{Example 2.} If\; $\displaystyle f(x,\,y) := \frac{x^2}{x^2+y^2}$,\, then
\begin{itemize}
\item $\lim_{x\to 0}\left(\lim_{y\to 0}f(x,\,y)\right) = \lim_{x\to0} \frac{x^2}{x^2} = 1$
\item $\lim_{y\to 0}\left(\lim_{x\to 0}f(x,\,y)\right) = \lim_{y\to0} 0 = 0$
\item the usual limit $\lim_{(x,y)\to(0,0)}f(x,\,y)$ again does not exist, \PMlinkescapetext{even} though both of the iterated limits do.
\end{itemize} |
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