Golden Ratio Limit

$L=\lim_{x\rightarrow 0} \Big(\frac{\sum_{n=1}^{1008}\phi^{nx}}{1008}\Big)^{1/x}\Rightarrow \ln{L}=\lim_{x\rightarrow 0} \frac{\ln{\sum_{n=1}^{1008}\phi^{nx}}-\ln{1008}}{x}$ Since both the numerator and denominator approach zero, as $x$ approaches zero, I can apply L'Hospital's rule. $\ln{L}=\lim_{x\rightarrow 0} \frac{\frac{\sum_{n=1}^{1008}n\phi^{nx}\ln{\phi}}{\sum_{n=1}^{1008}\phi^{nx}}}{1}$ Substituting $x=0$ , $\ln{L}=\frac{\sum_{n=1}^{1008}n\ln{\phi}}{1008}=\frac{\frac{(1008)(1009)}{2}\ln{\phi}}{1008}=\frac{1009}{2}\ln{\phi}$ $\Rightarrow L=\exp\Big(\frac{1009}{2}\ln{\phi}\Big)=\sqrt{\phi^{1009}}=\sqrt{F_{1009}\phi+F_{1008}}$