find the limit of the following sequence or determine that the sequence diverges. {n sin(11/n)} select the…

Explanation:

Step1: Let (x = \frac{1}{n})

As (n\rightarrow\infty), (x\rightarrow0). The sequence (\left{n\sin\left(\frac{11}{n}\right)\right}) can be rewritten as (\frac{\sin(11x)}{x}) where (x=\frac{1}{n}).

Step2: Use the limit - formula (\lim_{u\rightarrow0}\frac{\sin u}{u}=1)

We have (\lim_{n\rightarrow\infty}n\sin\left(\frac{11}{n}\right)=\lim_{x\rightarrow0}\frac{\sin(11x)}{x}). Multiply and divide by 11: (\lim_{x\rightarrow0}\frac{\sin(11x)}{x}=\lim_{x\rightarrow0}11\times\frac{\sin(11x)}{11x}). Let (u = 11x), as (x\rightarrow0), (u\rightarrow0). Then (\lim_{x\rightarrow0}11\times\frac{\sin(11x)}{11x}=11\lim_{u\rightarrow0}\frac{\sin u}{u}). Since (\lim_{u\rightarrow0}\frac{\sin u}{u}=1), we get (11\times1 = 11).

Answer:

A. The limit of the sequence is (11)