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

Problem 1.39 · hard

1. \[ \lim_{x \to \infty}\left(\frac{4 \left(3 x - 1\right)^{2} - 1}{3 x + 2 \left(3 x - 1\right)^{2} - 1}\right) \]
   limitStarting with the original limit expression.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{36 x^{2} - 24 x + 3}{18 x^{2} - 9 x + 1}\right) \]
   algebra algebra simplifyExpand the squared terms. Distribute the constants. Combine like terms in the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{36 - \frac{24}{x} + \frac{3}{x^{2}}}{18 - \frac{9}{x} + \frac{1}{x^{2}}}\right) \]
   algebra algebraDivide numerator and denominator by the highest power of x, which is x**2. Simplify the fractions.✓ Proved
4. \[ = \lim_{x \to \infty}\left(36 - \frac{24}{x} + \frac{3}{x^{2}}\right) \left(\lim_{x \to \infty}\left(18 - \frac{9}{x} + \frac{1}{x^{2}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = 2 \]
   limit simplifyEvaluate the limit as x approaches infinity, noting that terms with x in the denominator go to zero. Simplify the final fraction.✓ Proved