find the limit of the following sequence or determine that the sequence diverges.\n{-cos n/3n}\nselect the…

Explanation:

Step1: Recall the range of cosine function

We know that $- 1\leqslant\cos n\leqslant1$ for all $n\in N$. So, $-\frac{1}{3n}\leqslant-\frac{\cos n}{3n}\leqslant\frac{1}{3n}$.

Step2: Find the limits of the bounding - sequences

We find $\lim_{n\rightarrow\infty}-\frac{1}{3n}=0$ and $\lim_{n\rightarrow\infty}\frac{1}{3n}=0$.

Step3: Apply the Squeeze Theorem

Since $\lim_{n\rightarrow\infty}-\frac{1}{3n}=\lim_{n\rightarrow\infty}\frac{1}{3n} = 0$ and $-\frac{1}{3n}\leqslant-\frac{\cos n}{3n}\leqslant\frac{1}{3n}$, by the Squeeze Theorem, $\lim_{n\rightarrow\infty}-\frac{\cos n}{3n}=0$.

Answer:

A. The limit of the sequence is $0$.