Limit of \( \displaystyle \frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{2 \left(3 x - 1\right)^{3} + 5} \) as \( x \to \infty \)

Problem 1.328 · hard

1. \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{2 \left(3 x - 1\right)^{3} + 5}\right) \]
   limitStart with the limit of the given function as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{27 x^{3} - 27 x^{2} + 6 x}{54 x^{3} - 54 x^{2} + 18 x + 3}\right) \]
   simplifyExpand the binomial terms and simplify the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{27 - \frac{27}{x} + \frac{6}{x^{2}}}{54 - \frac{54}{x} + \frac{18}{x^{2}} + \frac{3}{x^{3}}}\right) \]
   algebra simplifyDivide both the numerator and the denominator by the highest power of x, which is x**3. Simplify the fractions within the limit.✓ Proved
4. \[ = \lim_{x \to \infty}\left(27 - \frac{27}{x} + \frac{6}{x^{2}}\right) \left(\lim_{x \to \infty}\left(54 - \frac{54}{x} + \frac{18}{x^{2}} + \frac{3}{x^{3}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = \frac{1}{2} \]
   limit simplifyEvaluate the limits of the numerator and denominator separately as x goes to infinity. Simplify the resulting fraction.✓ Proved