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

Problem 1.115 · medium

1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x - 1\right)^{3} + 1}{2 \left(x - 1\right)^{3} + 5}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} - 3 x^{2} + 2 x}{2 x^{3} - 6 x^{2} + 6 x + 3}\right) \]
   algebra simplifyExpand the cubic terms. Simplify the numerator and denominator by combining like terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{1 - \frac{3}{x} + \frac{2}{x^{2}}}{2 - \frac{6}{x} + \frac{6}{x^{2}} + \frac{3}{x^{3}}}\right) \]
   algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**3. Simplify each term in the expression.✓ Proved
4. \[ = \lim_{x \to \infty} \frac{1}{2} \]
   limit-lawApply the limit laws for each term as x approaches infinity.✓ Proved
5. \[ = \frac{1}{2} \]
   limitEvaluate the final numerical value.✓ Proved