Wavelets: How does the admissibility condition imply this $\Psi(0)=0$
It's obvious that Fourier transform of the mother wavelet is zero at the origin, otherwise the integral, whose argument is nonnegative, would be unbounded. Just because both the numerator and the denominator are zero, it does not mean it does not have a limit. So we know that $\Psi(0)=0$, but this is merely necessary, not sufficient for the integral to be bounded.

Consider the function $f: \omega \to |\sin(\omega)|^2/|\omega|$. How does this function's integral from $(-T,T)$ behave as $T \to \infty$. I hope that gives you some ideas.