Firstly, let's define the given sequence as:
(4n^2 + n + 2) / (9n^2 + 2n + 3)
To find the limit of this sequence as n approaches infinity, we divide the numerator and denominator by n^2. As n tends to infinity, the 'n' or constant terms become less significant, hence we can discard them.
When we divide the numerator by n^2, we get:
4 + 1/n + 2/n^2
When we divide the denominator by n^2, we get:
9 + 2/n + 3/n^2
When n tends to infinity, the terms 1/n and 2/n^2 in the numerator and the terms 2/n and 3/n^2 in the denominator become negligible, reaching towards zero as n increases, due to their inverse proportionality to n.
Hence, the sequence gets simplified to:
4/9
So, the limit of the given sequence as n approaches infinity is indeed 4/9.