Infinitely-large function

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:

$$\lim_{x\to x_0}f(x)=\infty.$$

The following are defined in a similar manner:

$$\lim_{x\to x_0\pm0}f(x)=\pm\infty,$$

$$\lim_{x\to\pm\infty}f(x)=\pm\infty.$$

For example,

$$\lim_{x\to-\infty}f(x)=+\infty$$

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.

Comments

See also Infinitesimal calculus.