Final answer:
The limit of the sequence xₙ= (sinⁿ t)/(1+ sin²ⁿ t) as n approaches infinity within the space X = C[0,3π] is 0 for all 't' excluding multiples of π.
Step-by-step explanation:
The question asks us to find the limit of the sequence xₙ= (sinⁿ t)/(1+ sin²ⁿ t) in the space X = C[0,3π].
Considering the properties of the sine function and its behavior in the given domain, we must take into account different scenarios based on the value of 't'. For values of 't' in [0, 3π], the sine function will oscillate between -1 and 1.
As 'n' goes to infinity, sinⁿ t will tend to 0 if |sin t| < 1, and will alternate between -1 and 1 if |sin t| = 1 (which happens at multiples of π), yielding either -1 or 1. Since |sin t| is never greater than 1, sinⁿ t will always tend to 0 for all t in [0, 3π] except for multiples of π.
Therefore, the sequence xₙ will have a limit of 0 for all 't' in [0, 3π] except for multiples of π. At multiples of π (where sin t = 0), the fraction is undefined, so we exclude these points when considering the limit of the sequence.