Question

1. -/1.11 points determine whether the sequence converges or diverges. if it converges, find the limit. (if the sequence diverges, enter diverges.) ( a _ { n } = \frac { n } { 4 } sin left( \frac { 4 } { n }

ight) ) ( lim _ { n
ightarrow infty } a _ { n } = )

Explanation:

Step1: Use the limit formula

We know that \(\lim_{x
ightarrow0}\frac{\sin x}{x} = 1\). Let \(x=\frac{4}{n}\), as \(n
ightarrow\infty\), \(x
ightarrow0\).

$$
LATEXBLOCK0
$$

Step2: Substitute the limit formula

Since \(\lim_{x
ightarrow0}\frac{\sin x}{x} = 1\), when \(x = \frac{4}{n}\) and \(n
ightarrow\infty\) (so \(x
ightarrow0\)), we have \(\lim_{n
ightarrow\infty}\frac{\sin\frac{4}{n}}{\frac{4}{n}}=1\)

Answer:

\(1\)