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

Problem 1.132 · medium

1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x + 1\right)^{3} - 1}{2 \left(x + 1\right)^{3} + 5}\right) \]
   limitEvaluate the limit of the given function as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} + 3 x^{2} + 2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
   simplifyExpand and simplify the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{x^{3}}{2 x^{3} + 6 x^{2} + 6 x + 7} + \frac{3 x^{2}}{2 x^{3} + 6 x^{2} + 6 x + 7} + \frac{2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
   limit-lawSplit the fraction into three separate parts using the limit law for sums.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) + \lim_{x \to \infty}\left(\frac{3 x^{2}}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) + \lim_{x \to \infty}\left(\frac{x^{3}}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
   limit-lawApply the limit law to each term individually.✓ Proved
5. \[ = \lim_{x \to \infty} \frac{1}{2 + \frac{6}{x} + \frac{6}{x^{2}} + \frac{7}{x^{3}}} + \lim_{x \to \infty}\left(\frac{3}{2 x + 6 + \frac{6}{x} + \frac{7}{x^{2}}}\right) + \lim_{x \to \infty}\left(\frac{2}{2 x^{2} + 6 x + 6 + \frac{7}{x}}\right) \]
   algebraDivide the numerator and denominator of each term by the highest power of x in the denominator.✓ Proved
6. \[ = \lim_{x \to \infty} \frac{1}{2 + \frac{6}{x} + \frac{6}{x^{2}} + \frac{7}{x^{3}}} \]
   limit-lawEvaluate the limits of the terms that approach zero.✓ Proved
7. \[ = \lim_{x \to \infty} \frac{1}{2} \]
   infinityEvaluate the limit as terms with 1/x approach zero.✓ Proved
8. \[ = \frac{1}{2} \]
   limitFinal evaluation of the limit.✓ Proved