Let

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

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

Evaluate the limit

\[\lim_{x \to 0} \left( \dfrac{f(x)+\tan \dfrac{x}{2^n}}{x} \right)^{1/x^2}\]

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