The original question is that I want to evaluate $$\lim \limits_{n \to \infty} ((n+1)\ln(n+1))^{\frac{1}{2}}-(n\ln(n))^{\frac{1}{2}}.$$
Now I want to find if for any $c \in (0,1)$, $$\lim \limits_{n \to \infty} ((n+1)\ln(n+1))^c-(n\ln(n))^c=0$$ holds. I prefer to evaluate it by definition, Cauchy sequence, or squeeze lemma. I don't want to use L'Hospital rule. I think I only need a hint.
For the original question, I have tried to represent $$((n+1)\ln(n+1))^{\frac{1}{2}}-(n\ln(n))^{\frac{1}{2}}$$ as $$(n\ln(n))^{\frac{1}{2}}(((1+\frac{1}{n})(1+\frac{\ln(1+\frac{1}{n})}{\ln(n)}))^{\frac{1}{2}}-1)$$ but lost insight to continue. That is a dull idea. But was it not $\frac{1}{2}$, this kind of idea would turn to be the only way I can think, again. Am I on the right way?