Questions: Evaluate the limit as n approaches infinity of (sqrt(16n^2 + 7n) - 4n). Enter your answer as a reduced fraction.

$Evaluate the limit as n approaches infinity of (sqrt(16n^2 + 7n) - 4n). Enter your answer as a reduced fraction.$

Transcript text: Evaluate $\lim _{n \rightarrow \infty}\left(\sqrt{16 n^{2}+7 n}-4 n\right)$. A graph of the first 20 terms of the sequence follows. Enter your answer as a reduced fraction.

Solution

Solution Steps

Step 1: Rewrite the expression

We can rewrite the expression inside the limit as follows:

√(16n² + 7n) - 4n = (√(16n² + 7n) - 4n) * (√(16n² + 7n) + 4n) / (√(16n² + 7n) + 4n) = (16n² + 7n - 16n²) / (√(16n² + 7n) + 4n) = (7n) / (√(16n² + 7n) + 4n)

Step 2: Divide by n

Divide both the numerator and denominator by n:

(7n/n) / (√(16n²+7n)/n + 4n/n) = 7 / (√(16 + 7/n) + 4)

Step 3: Evaluate the limit

As n approaches infinity, 7/n approaches 0. Therefore:

lim (n→∞) 7 / (√(16 + 7/n) + 4) = 7 / (√(16 + 0) + 4) = 7 / (4 + 4) = 7/8