Evaluation of the limit $\lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$

Then what is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ ?


When $q\to 1^-$, $\tau \to 0$, hence $-1/\tau \to i\infty$,
$$\begin{aligned}\frac{{{\phi ^5}{{(q)}_\infty }}}{{\phi {{({q^5})}_\infty }}} = \frac{{{\eta ^5}(\tau )}}{{\eta (5\tau )}}  &= \frac{{{{\left( {\frac{{ - 1}}{{\tau i}}} \right)}^{5/2}}{\eta ^5}(\frac{{ - 1}}{\tau })}}{{{{\left( {\frac{{ - 1}}{{5\tau i}}} \right)}^{1/2}}\eta (\frac{{ - 1}}{{5\tau }})}} \\ &= -\frac{{\sqrt 5 }}{{{\tau ^2}}}\frac{{{e^{ - \frac{{5\pi i}}{{12\tau }}}}\prod\limits_{n = 1}^\infty  {{{\left( {1 - {e^{ - \frac{{2\pi ni}}{\tau }}}} \right)}^5}} }}{{{e^{ - \frac{{\pi i}}{{60\tau }}}}\prod\limits_{n = 1}^\infty  {\left( {1 - {e^{ - \frac{{2\pi ni}}{{5\tau }}}}} \right)} }}\\ &\sim -\frac{{\sqrt 5 }}{{{\tau ^2}}}{e^{ - \frac{{2\pi i}}{{5\tau }}}} \to 0\end{aligned}$$
Note that the two infinite products both $\to 1$.
