Limit of \( \displaystyle \frac{27 x^{3} - 3 x}{54 x^{3} + 5} \) as \( x \to \infty \)

Problem 1.154 · medium

1. \[ \lim_{x \to \infty}\left(\frac{27 x^{3} - 3 x}{54 x^{3} + 5}\right) \]
   limit algebraIdentify the limit to be evaluated. Divide both numerator and denominator by x**3.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{27 - \frac{3}{x^{2}}}{54 + \frac{5}{x^{3}}}\right) \]
   simplifySimplify the terms inside the limit.✓ Proved
3. \[ = \lim_{x \to \infty}\left(27 - \frac{3}{x^{2}}\right) \left(\lim_{x \to \infty}\left(54 + \frac{5}{x^{3}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
4. \[ = \frac{1}{2} \]
   limit simplifyEvaluate the limits of the numerator and denominator separately. Reduce the fraction to its simplest form.✓ Proved