Final answer:
The limit of the sequence x₍ₙ = (2tⁿ - 1)/(1 + tⁿ) is 2 for t > 1 and 1/2 for t = 1 within the given space X = C[1,2]. There is no case for t < 1 in the given space.
Step-by-step explanation:
The question requires us to find the limit of the sequence x₍ₙ = (2tⁿ - 1)/(1 + tⁿ) as n approaches infinity in the space X = C[1,2]. We have to consider the behavior of tⁿ as n grows without bound. If t > 1, then tⁿ grows infinitely large, and the dominant term in both the numerator and denominator is tⁿ. Thus, the sequence approaches the ratio of the coefficients of the dominant terms, which would be 2/1 = 2. If t = 1, each term in the sequence is simply (2 - 1) / (1 + 1) = 1/2, which is a constant sequence and hence its limit is 1/2.
Therefore, the limit of the sequence x₍ₙ depends on the value of t. If t > 1, the limit is 2. If t = 1, the limit is 1/2. There is no t < 1 in the given space X = C[1,2].