I have found the this exercise in my book of introduction to calculus, could someone help me with it?
Let $N\geq 2$ be a natural number, prove that the following limit is equal to $0$: $$\displaystyle\lim_{t \to{0^+}}{t^{N-1}\left[\ln \left(\ln\left( 1+\frac{1}{t} \right) \right) \right]^N }.$$
I tried to use L’Hospital's rule but the successive derivatives has a very complicated expression. So, is there an easier way to solve it? Any help will be appreciated.