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

Problem 1.149 · medium

1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 1}{3 x - 2}\right) \]
   limit simplifyStart with the limit of the function as x approaches 2/3 from the right. The limit is the same from both sides since the function is continuous near 2/3 except at the point itself.✓ Proved
2. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x}{3 x - 2}\right) \]
   algebra simplifyExpand the squared term in the numerator. Simplify the numerator by subtracting the constants.✓ Proved
3. \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x\right) \]
   factor cancelFactor the numerator to reveal the common factor in the denominator. Cancel the common factor (3*x - 2) from the numerator and denominator.✓ Proved
4. \[ = 2 \]
   limit simplifyEvaluate the remaining expression at x = 2/3. Final result.✓ Proved