Final answer:
To find the limit of (2n^2 - 5n + 8) / 3(n - 2)^2 as n approaches infinity, we simplify the expression by dividing by n^2 and ignoring lower-power terms, resulting in a limit of 2/3.
Step-by-step explanation:
The question involves finding the limit of the function (2n^2 - 5n + 8) / 3(n - 2)^2 as n tends to infinity, using the algebra of limits. To answer this question, we'll consider the highest powers of n in the numerator and the denominator.
Firstly, we recognize that as n approaches infinity, the terms with lower powers of n become negligible compared to the highest power term. So, we can simplify the expression by dividing the numerator and the denominator by n^2, the highest power of n in the expression.
This simplification yields:
(2 - 5/n + 8/n^2) / (3/n^2)(1 - 4/n + 4/n^2)
Because the terms with n in the denominator tend to zero as n approaches infinity, we are left with:
(2/3)
Therefore, the limit of the given function as n approaches infinity is 2/3.