Limit of \( \displaystyle \frac{- 9 x + 2 \left(3 x + 1\right)^{2} - 5}{3 x - 1} \) as \( x \to \frac{1}{3} \)

Problem 1.372 · hard

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 9 x + 2 \left(3 x + 1\right)^{2} - 5}{3 x - 1}\right) \]
   limit algebraStart with the limit of the given function. Rearrange the terms in the numerator.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{18 x^{2} + 3 x - 3}{3 x - 1}\right) \]
   simplify simplify simplify factorExpand the squared binomial. Distribute the 2. Combine like terms in the numerator. Factor out the constant 3.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(2 x + 1\right) \left(9 x - 3\right)}{3 x - 1}\right) \]
   factorFactor the quadratic expression in the numerator.✓ Proved
4. \[ = \lim_{x \to \frac{1}{3}^+}\left(6 x + 3\right) \]
   cancelCancel the common factor (3*x - 1).✓ Proved
5. \[ = 5 \]
   limit algebra algebra simplifyEvaluate the limit by substituting x = 1/3. Multiply 2 by 1/3. Add the terms in the parentheses. Final simplification.✓ Proved