Question9. Calculate the value of the following limit: $$\lim _{n \rightarrow \infty}\left(\sin ^{n}(\pi / 6)+\sin ^{n}(\pi / 3)+\sin ^{n}(\pi / 2)\right)^{1 / n}$$

Studdy Solution
Apply the limit properties. The dominant term $1^n = 1$ simplifies the expression:
$$\lim_{n \to \infty} \left(\frac{1}{2}^n + \left(\frac{\sqrt{3}}{2}\right)^n + 1^n\right)^{1/n}$$
As $n \to \infty$, the terms $\frac{1}{2}^n$ and $\left(\frac{\sqrt{3}}{2}\right)^n$ approach 0, leaving:
$$\lim_{n \to \infty} \left(0 + 0 + 1\right)^{1/n} = 1^{1/n} = 1$$
The value of the limit is:
$$\boxed{1}$$