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To evaluate the limit lim⁡n→∞12+32e−2i(n+1)x12+32e−2inx,\lim_{n \to \infty} \frac{\frac{1}{2} + \frac{3}{2} e^{-2i(n+1)x}}{\frac{1}{2} + \frac{3}{2} e^{-2inx}},n→∞lim21+23e−2inx21+23e−2i(n+1)x, we need to analyze the behavior of the numerator and the denominator as n→∞n \to \inftyn→∞. First...