Logarithm as limit

$\ln z$ is the derivative of $t\mapsto z^t$ at $t=0$, so $$\ln z = \lim_{h\to 0}\frac{ z^h-1}h=\lim_{\omega\to \infty} \omega(z^{1/\omega}-1).$$

You have $z^{1/\omega}= \exp ( \ln(z)/\omega)= 1+ \ln(z)/\omega + o(1/\omega)$, so $\ln(z)=\lim\limits_{\omega \to + \infty} \omega (z^{1/\omega}-1)$.

Another way is to use L'Hôpital's rule. Write $w(z^{1/w}-1)$ as $$\frac{z^{1/w}-1}{1/w}\,.$$ Note that both the numerator and the denominator go to 0 as $w\to\infty$; thus, we can apply L'Hôpital's rule: $$\lim_{w\to\infty}\frac{z^{1/w}-1}{1/w} =\lim_{w\to\infty}\frac{\frac{d}{dw}(z^{1/w}-1)}{\frac{d}{dw}1/w}=\lim_{w\to\infty}\frac{(-1/w^{2})(\ln z)z^{1/w}}{-1/w^2}=\ln z \,. $$ In the numerator, we used $z^{1/w}=e^{(1/w)\ln z}$ and then proceeded as follows: $$ \frac{d}{dw}e^{(1/w)\ln z}=\left[\frac{d}{dw}\Big((1/w)\ln z\Big)\right]e^{(1/w)\ln z}=\left[(-1/w^{2})\ln z\right]z^{1/w}\,. $$