improper limits ImproperLimits 2004-10-03 04:01:51 2008-01-14 14:31:46 Definition limit mnemonic of infinite % 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 In calculus there is often used such expressions as ``the limit of a function is infinite'', and one may write for instance that $$\lim_{x \to 0}\frac{1}{x^2} = \infty.$$ Such ``limits'' are actually \PMlinkescapetext{extensions} of the limit notion, and can be defined exactly.\, They are called {\em improper limits}. \textbf{Definition.}\, Let the real function $f$ be defined in a neighbourhood of the point $x_0$. $$\lim_{x \to x_0}f(x) = \infty$$ iff for every real number $M$ there exists a number $\delta_M$ such that $$f(x) > M$$ as soon as $$0 < |x-x_0| < \delta_M.$$ In a similar way we can define the improper limit $-\infty$ of a real function.\, The definition may be extended also to the cases\, $x \to \pm\infty$.\\ \textbf{Note 1.}\, If\, $\lim_{x \to x_0}f(x) = \infty$\, and\, $\lim_{x \to x_0}g(x) = a > 0$,\, then we have $$\lim_{x \to x_0}f(x)g(x) = \infty.$$ Hence we can say that\, $\infty\cdot a = \infty$\, when\, $a > 0$.\, There are some other \PMlinkescapetext{comparable} ``mnemonics of infinite'' (cf. the extended real numbers): $$\infty\cdot a = -\infty \quad(a < 0)$$ $$\pm\infty+a = \pm\infty$$ $$\frac{a}{\pm\infty} = 0$$ $$\infty+\infty = \infty$$ $$\infty\cdot\infty = \infty$$ $$-\infty\cdot\infty = -\infty$$ On the contrary, there exist no mnemonics for the cases $$\infty\cdot0,\,\, \infty-\infty,\,\, \frac{\infty}{\infty},\,\, \frac{0}{0},\,\, 0^0,\,\, \infty^0,\,\, 1^\infty;$$ they are \PMlinkescapetext{indefinite} and depend on the instance (cf. the indeterminate form).\\ \textbf{Note 2.}\, In the complex plane, the expression $$\lim_{z \to z_0}f(z) = \infty$$ means that\, $\displaystyle \lim_{z \to z_0}|f(z)| = \infty$.