Leaps and bounds

Calculus Level 3

Let $$x_n$$ be a positive root of the equation $$x^n = x^2 + x + 1.$$

Find $$\displaystyle \exp\Big(\lim_{n \to \infty} n\left(x_n - 1\right)\Big).$$


Notation: $$\exp( a ) = e^a$$

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