Infinities and Infinitesimals: Definitions, Comparison and Asymptotic Equivalence

The meaning of infinite and infinitesimal

The concepts of infinite and infinitesimal describe the behaviour of a function with respect to a given limiting process, for instance as \(x\to x_0\) or as \(x\to \pm\infty\).

To say that a function is infinite means that its values tend to \(+\infty\) or to \(-\infty\). The symbol \(\infty\) therefore does not represent a real number, but expresses unbounded behaviour.

To say instead that a function is infinitesimal means that its values tend to zero. Here too, what is being described is not the value taken by the function at a single point, but its behaviour throughout the limiting process under consideration.

The same function may be infinite with respect to one limiting process and infinitesimal with respect to another. For this reason, every statement must specify the limiting process to which it refers.

Infinite functions

Let \(f\) be a function defined in a neighbourhood of \(x_0\), possibly excluding the point \(x_0\) itself. We say that \(f\) is infinite as \(x\to x_0\) if

\[ \lim_{x\to x_0}f(x)=+\infty \]

or

\[ \lim_{x\to x_0}f(x)=-\infty. \]

In the first case, the values of \(f(x)\) become greater than any preassigned real number; in the second, they become smaller than any preassigned real number.

For example, the function

\[ f(x)=\frac{1}{x^2} \]

is infinite as \(x\to 0\), since

\[ \lim_{x\to 0}\frac{1}{x^2}=+\infty. \]

Likewise, a function may be infinite as \(x\to+\infty\) or as \(x\to-\infty\). For example,

\[ \lim_{x\to+\infty}x^2=+\infty. \]

The expression "infinite function" is therefore meaningful only in relation to a specific limiting process.

Infinitesimal functions

Let \(f\) be a function defined in a neighbourhood of \(x_0\), possibly excluding the point \(x_0\) itself. We say that \(f\) is infinitesimal as \(x\to x_0\) if

\[ \lim_{x\to x_0}f(x)=0. \]

This means that the values of \(f(x)\) can be made arbitrarily small in absolute value, provided \(x\) is sufficiently close to \(x_0\).

For example, the function

\[ f(x)=x^2 \]

is infinitesimal as \(x\to 0\), since

\[ \lim_{x\to 0}x^2=0. \]

A function may also be infinitesimal as \(x\to+\infty\) or as \(x\to-\infty\). For example,

\[ \lim_{x\to+\infty}\frac{1}{x}=0, \]

so the function \(\displaystyle\frac{1}{x}\) is infinitesimal as \(x\to+\infty\).

Here too, the expression "infinitesimal function" must always be understood with reference to a specific limiting process.

Comparison of infinities

Let \(f\) and \(g\) be two infinite functions with respect to the same limiting process. To compare the rate at which they grow in absolute value, one considers the limit of their ratio:

\[ \lim\frac{f(x)}{g(x)}. \]

If

\[ \lim\frac{f(x)}{g(x)}=0, \]

then \(f\) is an infinity of lower order than \(g\): the absolute value of \(g(x)\) grows more rapidly than that of \(f(x)\).

If instead

\[ \lim\left|\frac{f(x)}{g(x)}\right|=+\infty, \]

then \(f\) is an infinity of higher order than \(g\).

Finally, if

\[ \lim\frac{f(x)}{g(x)}=L, \qquad 0<|L|<+\infty, \]

the two functions are infinities of the same order.

For example, as \(x\to+\infty\),

\[ \lim_{x\to+\infty}\frac{x}{x^2} = \lim_{x\to+\infty}\frac{1}{x} =0. \]

Hence \(x\) is an infinity of lower order than \(x^2\).

If none of the three preceding cases occurs, the two functions cannot be compared by means of this classification.

Comparison of infinitesimals

Let \(f\) and \(g\) be two infinitesimal functions with respect to the same limiting process, with \(g(x)\neq 0\) eventually. Here too, the comparison is carried out by studying the limit of the ratio

\[ \lim\frac{f(x)}{g(x)}. \]

If

\[ \lim\frac{f(x)}{g(x)}=0, \]

then \(f\) is an infinitesimal of higher order than \(g\): the function \(f\) tends to zero more rapidly than \(g\).

If instead

\[ \lim\left|\frac{f(x)}{g(x)}\right|=+\infty, \]

then \(f\) is an infinitesimal of lower order than \(g\).

Finally, if

\[ \lim\frac{f(x)}{g(x)}=L, \qquad 0<|L|<+\infty, \]

the two functions are infinitesimals of the same order.

For example, as \(x\to 0\),

\[ \lim_{x\to 0}\frac{x^2}{x} = \lim_{x\to 0}x =0. \]

Hence \(x^2\) is an infinitesimal of higher order than \(x\).

If none of the three preceding cases occurs, the two functions cannot be compared by means of this classification.

Order of an infinity and order of an infinitesimal

Comparison by means of the ratio allows one to specify more precisely the rate at which a function tends to zero, or to \(+\infty\) or to \(-\infty\).

Let \(f\) and \(g\) be two infinitesimals with respect to the same limiting process, with \(g(x)>0\) eventually. We say that \(f\) is an infinitesimal of order \(\alpha>0\) with respect to \(g\) if

\[ \lim\frac{f(x)}{[g(x)]^\alpha}=L, \qquad 0<|L|<+\infty. \]

For example, as \(x\to 0^+\), the function \(x^3\) is an infinitesimal of order \(3\) with respect to \(x\), since

\[ \lim_{x\to 0^+}\frac{x^3}{x^3}=1. \]

Likewise, let \(f\) and \(g\) be two infinities with respect to the same limiting process, with \(g(x)>0\) eventually. We say that \(f\) is an infinity of order \(\alpha>0\) with respect to \(g\) if

\[ \lim\frac{f(x)}{[g(x)]^\alpha}=L, \qquad 0<|L|<+\infty. \]

For example, as \(x\to+\infty\), the function \(x^4\) is an infinity of order \(4\) with respect to \(x\), because

\[ \lim_{x\to+\infty}\frac{x^4}{x^4}=1. \]

The order is therefore not an absolute property of the function, but depends on the function chosen as the term of comparison.

Equivalent infinities and infinitesimals

Let \(f\) and \(g\) be two functions, either both infinite or both infinitesimal, with respect to the same limiting process, with \(g(x)\neq 0\) eventually. The two functions are said to be asymptotically equivalent if

\[ \lim\frac{f(x)}{g(x)}=1. \]

In this case one writes

\[ f(x)\sim g(x). \]

Asymptotic equivalence expresses the fact that, in the limiting process under consideration, the ratio of the two functions tends to \(1\): they therefore exhibit the same principal behaviour.

For example, as \(x\to 0\),

\[ \lim_{x\to 0}\frac{\sin x}{x}=1, \]

so that

\[ \sin x\sim x. \]

Likewise, as \(x\to+\infty\),

\[ \lim_{x\to+\infty}\frac{x^2+x}{x^2} = \lim_{x\to+\infty}\left(1+\frac{1}{x}\right) =1, \]

so that

\[ x^2+x\sim x^2. \]

Two equivalent infinities or infinitesimals are of the same order. The converse, however, need not hold: if the ratio tends to a finite, non-zero constant different from \(1\), the functions have the same order but are not equivalent.

Using equivalents in the calculation of limits

Asymptotic equivalences make it possible to replace a function with a simpler one having the same principal behaviour in the limiting process under consideration.

If

\[ f(x)\sim g(x), \]

then, for the purpose of evaluating the limit, \(f(x)\) may be replaced by \(g(x)\) when it occurs as a factor in a product or quotient, provided the resulting expressions are eventually well defined.

For example, since

\[ \sin x\sim x \qquad\text{as }x\to 0, \]

we have

\[ \lim_{x\to 0}\frac{\sin x}{x}=1. \]

Likewise, using

\[ 1-\cos x\sim\frac{x^2}{2} \qquad\text{as }x\to 0, \]

we obtain

\[ \lim_{x\to 0}\frac{1-\cos x}{x^2} = \frac{1}{2}. \]

Substitution by means of equivalents cannot, however, be applied automatically in sums and differences, since cancellation of the leading terms may occur.

For example, as \(x\to+\infty\),

\[ x+1\sim x, \]

but replacing \(x+1\) with \(x\) in the difference

\[ (x+1)-x \]

would incorrectly lead to \(0\), whereas the expression is identically equal to \(1\).

Equivalents must therefore be used with due regard to the structure of the expression: they are particularly effective in products and quotients, whereas in sums and differences one must first identify the dominant term, or transform the expression appropriately.

Relationship between infinities and infinitesimals

Infinities and infinitesimals are related through the operation of taking reciprocals. If \(f(x)\) is infinite, then it is eventually different from zero, and

\[ \lim\frac{1}{f(x)}=0. \]

Conversely, if \(f(x)\) is infinitesimal and eventually different from zero, then

\[ \lim\left|\frac{1}{f(x)}\right|=+\infty. \]

If \(f(x)>0\) eventually, its reciprocal tends to \(+\infty\); if instead \(f(x)<0\) eventually, its reciprocal tends to \(-\infty\). If \(f(x)\) is not eventually of constant sign, its reciprocal tends neither to \(+\infty\) nor to \(-\infty\), although its absolute value tends to \(+\infty\).

For example, as \(x\to 0\),

\[ x^2\to 0 \qquad\text{and}\qquad \frac{1}{x^2}\to+\infty. \]

Instead, as \(x\to 0^+\),

\[ x\to 0^+ \qquad\text{and}\qquad \frac{1}{x}\to+\infty, \]

while as \(x\to 0^-\),

\[ x\to 0^- \qquad\text{and}\qquad \frac{1}{x}\to-\infty. \]

Principal asymptotic equivalences

In the calculation of limits, certain fundamental asymptotic equivalences are frequently used. With angles expressed in radians, the following equivalences hold as \(x\to 0\):

\[ \sin x\sim x, \qquad \tan x\sim x, \qquad \arcsin x\sim x, \qquad \arctan x\sim x; \]

\[ 1-\cos x\sim\frac{x^2}{2}; \]

\[ e^x-1\sim x, \qquad \ln(1+x)\sim x; \]

\[ (1+x)^\alpha-1\sim\alpha x, \qquad \alpha\in\mathbb{R},\ \alpha\neq 0. \]

These equivalences follow from standard limits and allow more complicated infinitesimals to be replaced by simpler expressions when they occur as factors in products or quotients.

For example, since

\[ e^x-1\sim x \qquad\text{and}\qquad \sin x\sim x \qquad\text{as }x\to 0, \]

one obtains

\[ \lim_{x\to 0}\frac{e^x-1}{\sin x} = \lim_{x\to 0}\frac{x}{x} =1. \]

Fundamental hierarchies among infinities and infinitesimals

As \(x\to+\infty\), the principal families of functions display different rates of growth. In particular, for every \(\alpha>0\) and every \(a>1\),

\[ \lim_{x\to+\infty}\frac{\ln(x)}{x^\alpha}=0 \qquad\text{and}\qquad \lim_{x\to+\infty}\frac{x^\alpha}{a^x}=0. \]

This means that every positive power of \(x\) grows more rapidly than the logarithm, while every exponential function with base greater than \(1\) grows more rapidly than any positive power of \(x\).

To express this hierarchy rigorously, once the limiting process is fixed and assuming \(g(x)\neq 0\) eventually, the notation

\[ f(x)=o(g(x)) \]

means that

\[ \lim\frac{f(x)}{g(x)}=0. \]

Hence, as \(x\to+\infty\),

\[ \ln(x)=o(x^\alpha) \qquad\text{and}\qquad x^\alpha=o(a^x). \]

For example,

\[ \lim_{x\to+\infty}\frac{\ln(x)}{\sqrt{x}}=0 \qquad\text{and}\qquad \lim_{x\to+\infty}\frac{x^{10}}{2^x}=0. \]

Passing to reciprocals yields the corresponding hierarchy among infinitesimals:

\[ \frac{1}{a^x}=o\left(\frac{1}{x^\alpha}\right) \qquad\text{and}\qquad \frac{1}{x^\alpha}=o\left(\frac{1}{\ln(x)}\right) \qquad\text{as }x\to+\infty. \]

In each relation, the function on the left tends to zero more rapidly, and is therefore an infinitesimal of higher order than the one on the right.

Conclusions

The concepts of infinite and infinitesimal make it possible to describe and compare the principal asymptotic behaviours of functions. An infinite function tends to \(+\infty\) or to \(-\infty\), while an infinitesimal function tends to zero; in both cases, the behaviour must always be referred to a specific limiting process.

Studying the ratio of two functions makes it possible to compare their behaviour: depending on the limit obtained, one can establish which of the two prevails, whether they are of the same order, or whether they are asymptotically equivalent. Equivalences are particularly useful in the calculation of limits, provided they are applied correctly and without arbitrary substitutions in sums and differences.

Infinities and infinitesimals thus provide an essential framework for identifying the dominant terms of an expression and rigorously simplifying the study of limits.