Limit of \( \displaystyle \frac{27 x^{3} - 1}{9 x^{2} - 1} \) as \( x \to \frac{1}{3} \)

Problem 1.310 · medium

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{27 x^{3} - 1}{9 x^{2} - 1}\right) \]
   limitEvaluate the limit of the given function as x approaches 1/3.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(3 x - 1\right) \left(9 x^{2} + 3 x + 1\right)}{9 x^{2} - 1}\right) \]
   factorFactor the numerator using the difference of cubes formula.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 3 x + 1}{3 x + 1}\right) \]
   factor cancelFactor the denominator using the difference of squares formula. Cancel the common factor (3*x - 1).✓ Proved
4. \[ = \lim_{x \to \frac{1}{3}^+}\left(9 x^{2} + 3 x + 1\right) \left(\lim_{x \to \frac{1}{3}^+}\left(3 x + 1\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = \frac{3}{2} \]
   substitution simplify simplifySubstitute x = 1/3 into the expression. Simplify the arithmetic terms. Calculate the final value.✓ Proved