real analysis - How to solve this limit $limlimits_{nrightarrowinfty},{{left( frac{{{u}_{n}}}{{{v}_{n}}} right)}^{2040}}$?

Let $(u_n)_n$ and $(v_n)_n$ be positive numerical sequences such that $(1+\sqrt{2})^n=u_n+v_n\sqrt{2}$ with $n\in\mathbb{N}$

How to determine the limit of

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{u}_{n}}}{{{v}_{n}}} \right)}^{2040}}$$

I start think to find such recessive sequence for $u_n$ terms in order to separate them using induction but this leads to nothing . so as result is there any shortcut to attack this problem?