Finding the limit of $mathbb{E}[theta^n]/mathbb{E}[theta^{n-1}]$

Let $theta$ denote a smoothly distributed random variable with support $[0, 1]$. I am trying to evaluate

$$ lim_{n rightarrow infty} frac{mathbb{E}[theta^n]}{mathbb{E}[theta^{n-1}]}$$

I suspect, but cannot show, that the limit equals $1$. Does anyone know how to do this?

My attempts so far: Since $theta in [0, 1]$, it seems reasonably clear that both $mathbb{E}[theta^n] rightarrow 0$ and $mathbb{E}[theta^{n-1}] rightarrow 0$ as $n rightarrow infty$ (we are raising numbers that are less than $1$ to ever higher powers). Thus, we can apply L’Hopital’s rule to find that

$$ lim_{n rightarrow infty} frac{mathbb{E}[theta^n]}{mathbb{E}[theta^{n-1}]} equiv lim_{n rightarrow infty} frac{int_0^1 theta^nf(theta)dtheta}{int_0^1 theta^{n-1}f(theta)dtheta} = lim_{n rightarrow infty} frac{int_0^1 ln(theta)theta^nf(theta)dtheta}{int_0^1 ln(theta)theta^{n-1}f(theta)dtheta}$$

I am a bit unclear, however, how to proceed from this point (or whether better approaches are available).