\documentclass[a5paper]{book}
\usepackage{amsmath}

\makeindex

\begin{document}
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\noindent \tiny tiny
\scriptsize scriptsize
\footnotesize footnotesize
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\noindent
scriptscriptstyle: $\scriptscriptstyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n$ \\
scriptstyle: $\scriptstyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n$ \\
textstyle: $\textstyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n$ \\
displaystyle: $\displaystyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n$ \\

\end{document}