$$ \lim_{\theta\to0} \theta^{\frac1x -1} \tan(\theta^{\frac1x}) \;\;\;\;\; (x > 1) $$
I've tried L'Hôpital's rule with $\theta$ in the denominator, but successive applications seems to only lead to more complex expressions. Interestingly, it seems that each application of L'Hôpital's rule will produce another limit to which L'Hôpital's is applicable.
Can I use this fact somehow to analytically evaluate the limit?