calculus - Evaluation of $lim_{nrightarrow infty}left{left(2+sqrt{3}right)^{2n}right};,$ Where $nin mathbb{N}.$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\left\{\left(2+\sqrt{3}\right)^{2n}\right\}\;,$ Where $n\in \mathbb{N}.$

$\bf{My\; Try::}$ Let

$$\left(2+\sqrt{3}\right)^{2n} = I +f\;,$$ where $0

Now Let

$$0<\left(2-\sqrt{3}\right)^{2n}<1\;,$$ So $$\left(2-\sqrt{3}\right)^{2n}=f'$$ .

So

$$\left(2+\sqrt{3}\right)^{2n}+\left(2-\sqrt{3}\right)^{2n} = I +f+f' = \bf{Integer\; Quantity.}$$

So

$$f+f'\in \mathbb{Z}.$$ Now Given $0<(f+f')<2$. So $f+f' = 1\in \mathbb{Z}$

So

$$I+f+f' = \bf{Integer\; Quantity}\Rightarrow f = Integer\; Quantity-f'$$

So

$$\displaystyle \lim_{n\rightarrow \infty}\left\{\left(2+\sqrt{3}\right)^{2n}\right\} = \bf{Integer\; quantity-}\displaystyle \lim_{n\rightarrow \infty}\left\{\left(2-\sqrt{3}\right)^{2n}\right\}$$

Now how can i solve after that, help me, Thanks