Which of the following assertions is correct?

Correct option is C

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$$\lim_{n \to \infty} e^{\sin\left(\frac{n\pi + (-1)^n 2e}{2n}\right)} \\= \lim_{n \to \infty} e^{\sin\left(\frac{\pi + \frac{(-1)^n 2e}{n}}{2}\right)}= e^{\sin\left(\frac{\pi}{2}\right)} = e\\\therefore \limsup e^{\sin\left(\frac{n\pi + (-1)^n 2e}{2n}\right)} = e \\e \approx 2.718 \quad (\text{approx.})\quad \text{and} \quad \pi \approx 3.14 \quad (\text{approx.})\\\therefore e < \pi \implies \limsup e^{\sin\left(\frac{n\pi + (-1)^n 2e}{2n}\right)} < \pi$$

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