\chapter{An Elementary Example with Domain Restriction}

This chapter illustrates that the notion of an asymptotic
cannot be separated from the specification of the domain
in which the limiting process is considered.

Even for elementary functions,
different choices of the domain may lead
to different asymptotic interpretations.

\section{The Function}

Consider the real-valued function
\[
f(x)=
\begin{cases}
x^2 \sin\!\left(\frac{1}{x}\right), & x \neq 0, \\[4pt]
0, & x = 0 .
\end{cases}
\]

This function is continuous at \(x=0\),
but it oscillates infinitely often
in any punctured neighborhood of the origin.

\section{Naive Asymptotic Question}

A natural question is:

\begin{quote}
Can the constant \(0\) be interpreted as an asymptotic
of \(f(x)\) as \(x \to 0\)?
\end{quote}

The answer depends on what is meant by
``\(x \to 0\)'', that is, on the set of values
of \(x\) that are allowed in the limiting process.

\section{Derivative and Asymptotic}

A similar ambiguity appears for the derivative.
The derivative of \(f\) does not have a limit at \(x=0\),
and therefore \(0\) cannot be interpreted
as an asymptotic of \(f'(x)\) in any neighborhood
of the origin on the real axis.

This already indicates that asymptotic behavior
cannot be characterized solely by local expressions;
the admissible range of the variable is essential.

\section{Complex Extension}

Extend the function to the complex plane by
\[
f(z) = z^2 \sin\!\left(\frac{1}{z}\right),
\qquad z \in \mathbb{C}, \ z \neq 0 ,
\]
with \(f(0)=0\).

In the complex plane,
the question of asymptotic behavior
necessarily involves the choice of a domain
in which the limit is taken.

\section{Restricted Domain}

Consider the domain
\[
D = \{\, z \in \mathbb{C} : \Im(z) < \Re(z)^2 \,\}.
\]

Within this domain,
the function \(f(z)\) satisfies
\[
\lim_{z \to 0,\ z \in D} \frac{f(z)}{z^2} = 0 .
\]

In this restricted sense,
the constant \(0\) can be interpreted
as an asymptotic of \(f(z)\) at \(z \to 0\)
within the domain \(D\).

\section{Interpretation}

This example demonstrates that:

\begin{itemize}
\item An asymptotic relation requires
      the specification of a domain.
\item Without such a restriction,
      the notion of asymptotic behavior
      may become ambiguous or meaningless.
\item Even elementary functions
      may admit different asymptotic interpretations
      in different domains.
\end{itemize}

The figure below illustrates the chosen domain
and the behavior of the function in the complex plane.

