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For a positive integer n, let f(n) be defined as n plus the sum of terms in the form (a + bn − cn²) / (dn + en²), where the coefficients vary across terms. Specifically, f(n) = n + (16 + 5n − 3n²) / (4n + 3n²) + (32 + n − 3n²) / (8n + 3n²) + (48 − 3n − 3n²) / (12n + 3n²) +... + (25n − 7n²) / (7n²). Determine the value of lim n→∞ f(n).

1. 3 + (4/3) logₑ 7
2. 4 − (3/4) logₑ (7/3)
3. 4 − (4/3) logₑ (7/3)
4. 3 + (3/4) logₑ 7

Correct answer: 4 − (3/4) logₑ (7/3)

Solution

As n approaches infinity, the dominant terms in the function f(n) are analyzed. The logarithmic terms and coefficients simplify to yield the limit value of 4 − (3/4) logₑ (7/3).

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