\[f(x)=\tan\left( \frac{x}{2} \right) \cdot \sec \left( x \right)+\tan\left( \frac{x}{2^2} \right) \cdot \sec \left( \frac{x}{2} \right)+\ldots+\tan\left( \frac{x}{2^n} \right) \cdot \sec \left( \frac{x}{2^{n-1}} \right)\] \[ g(x)=f(x)+\tan\left( \frac{x}{2^n} \right) \]

where \(x \in \left( -\frac{\pi}{2} , \frac{\pi}{2} \right) \) and \(n \in \mathbb{N}\)

Evaluate the limit : \( \displaystyle \lim_{x \to 0} \left( \dfrac{g(x)}{x} \right)^{\frac{1}{x^3}}\)

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