To determine the limit of the sequence A(n) = n/5 * sin(4/n) as n approaches infinity, we can use the limit laws and theorems.
First, let's simplify the expression:
A(n) = (n/5) * sin(4/n)
As n approaches infinity, we have:
lim(n→∞) (n/5) = ∞/5 = ∞
Now, let's examine the term sin(4/n) as n approaches infinity:
lim(n→∞) sin(4/n) = sin(0) = 0
Using the limit laws, we can now find the limit of the sequence:
lim(n→∞) A(n) = lim(n→∞) [(n/5) * sin(4/n)]
= lim(n→∞) (n/5) * lim(n→∞) sin(4/n)
= ∞ * 0
= 0
Therefore, the limit of the sequence A(n) = n/5 * sin(4/n) as n approaches infinity is 0.