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f(x)=tan(x2)⋅sec(x)+tan(x22)⋅sec(x2)+…+tan(x2n)⋅sec(x2n−1)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)f(x)=tan(2x)⋅sec(x)+tan(22x)⋅sec(2x)+…+tan(2nx)⋅sec(2n−1x) g(x)=f(x)+tan(x2n) g(x)=f(x)+\tan\left( \frac{x}{2^n} \right) g(x)=f(x)+tan(2nx)
where x∈(−π2,π2)x \in \left( -\frac{\pi}{2} , \frac{\pi}{2} \right) x∈(−2π,2π) and n∈Nn \in \mathbb{N}n∈N
Evaluate the limit : limx→0(g(x)x)1x3 \displaystyle \lim_{x \to 0} \left( \dfrac{g(x)}{x} \right)^{\frac{1}{x^3}}x→0lim(xg(x))x31
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