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

Problem 1.369 · hard

1. \[ \lim_{x \to \infty}\left(\frac{3 \left(3 x + 1\right)^{2} + 1}{\left(3 x + 1\right)^{2} - 4}\right) \]
   limitEvaluate the limit as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{27 x^{2} + 18 x + 4}{9 x^{2} + 6 x - 3}\right) \]
   algebra simplifyExpand the squared terms. Simplify the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{27 + \frac{18}{x} + \frac{4}{x^{2}}}{9 + \frac{6}{x} - \frac{3}{x^{2}}}\right) \]
   algebra simplifyDivide numerator and denominator by x**2. Simplify the terms inside the limit.✓ Proved
4. \[ = \lim_{x \to \infty}\left(27 + \frac{18}{x} + \frac{4}{x^{2}}\right) \left(\lim_{x \to \infty}\left(9 + \frac{6}{x} - \frac{3}{x^{2}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = \frac{\lim_{x \to \infty}\left(27 + \frac{18}{x} + \frac{4}{x^{2}}\right)}{9} \]
   limit-lawEvaluate the limit of the denominator.✓ Proved
6. \[ = 3 \]
   limit simplifyEvaluate the limit of the numerator. Final simplification.✓ Proved