Limit of \( \displaystyle \frac{3 \left(2 x + 1\right)^{2} + 1}{\left(2 x + 1\right)^{2} - 4} \) as \( x \to \infty \)

Problem 1.368 · hard

1. \[ \lim_{x \to \infty}\left(\frac{3 \left(2 x + 1\right)^{2} + 1}{\left(2 x + 1\right)^{2} - 4}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{12 x^{2} + 12 x + 4}{4 x^{2} + 4 x - 3}\right) \]
   simplify simplifyExpand the squared terms. Distribute the 3 and combine like terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{12 + \frac{12}{x} + \frac{4}{x^{2}}}{4 + \frac{4}{x} - \frac{3}{x^{2}}}\right) \]
   algebra simplifyDivide numerator and denominator by x**2. Simplify the fractions inside the limit.✓ Proved
4. \[ = \lim_{x \to \infty}\left(12 + \frac{12}{x} + \frac{4}{x^{2}}\right) \left(\lim_{x \to \infty}\left(4 + \frac{4}{x} - \frac{3}{x^{2}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = \frac{\lim_{x \to \infty}\left(12 + \frac{12}{x} + \frac{4}{x^{2}}\right)}{4} \]
   limitEvaluate the limit of the denominator.✓ Proved
6. \[ = 3 \]
   limit simplify simplifyEvaluate the limit of the numerator. Simplify the resulting fraction. Final result.✓ Proved