A function of a variable $x$ whose absolute value becomes and remains larger than any given number as a result of variation of $x$. More exactly, a function $f$ defined in a neighbourhood of a point $x_0$ is called an infinitely-large function as $x$ tends to $x_0$ if for any number $M>0$ it is possible to find a number $\delta=\delta(M)>0$ such that for all $x\neq x_0$ satisfying $|x-x_0|<\delta$ the inequality $|f(x)|>M$ holds. This fact may be written as follows:
The following are defined in a similar manner:
means that for any $M>0$ it is possible to find a $\delta=\delta(M)>0$ such that the inequality $f(x)>M$ is valid for all $x<-\delta$. The study of infinitely-large functions may be reduced to that of infinitely-small functions (cf. Infinitely-small function), since $\psi(x)=1/f(x)$ will be infinitely small.
See also Infinitesimal calculus.
Infinitely-large function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Infinitely-large_function&oldid=33180