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

Problem 1.92 · hard

1. \[ \lim_{x \to 1^+}\left(\frac{- 9 x + 2 \left(3 x - 1\right)^{2} + 1}{3 x - 3}\right) \]
   limit factorIdentify the limit to evaluate. Factor the denominator.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{18 x^{2} - 21 x + 3}{3 x - 3}\right) \]
   simplify simplify simplify factorExpand the squared term. Distribute the constant 2. Combine like terms in the numerator. Factor out the common factor 3 from the numerator.✓ Proved
3. \[ = \lim_{x \to 1^+}\left(\frac{6 x^{2} - 7 x + 1}{x - 1}\right) \]
   simplifyCancel the common factor 3.✓ Proved
4. \[ = \lim_{x \to 1^+}\left(6 x - 1\right) \]
   factor cancelFactor the quadratic expression in the numerator. Cancel the common factor (x - 1).✓ Proved
5. \[ = 5 \]
   limitEvaluate the remaining limit.✓ Proved