Limit of \( \displaystyle \frac{18 x^{2} - 9 x - 2}{3 x - 2} \) as \( x \to \frac{2}{3} \)

Problem 1.43 · medium

1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{18 x^{2} - 9 x - 2}{3 x - 2}\right) \]
   limitEvaluate the limit of the given function at x = 2/3.✓ Proved
2. \[ = \lim_{x \to \frac{2}{3}^+} \text{NaN} \]
   simplifyNotice that the numerator and denominator both evaluate to zero at x = 2/3.Reviewed
3. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{\frac{d}{d x} \left(18 x^{2} - 9 x - 2\right)}{\frac{d}{d x} \left(3 x - 2\right)}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is in indeterminate form 0/0.Reviewed
4. \[ = \lim_{x \to \frac{2}{3}^+}\left(12 x - 3\right) \]
   simplify algebraCompute the derivatives of the numerator and the denominator. Simplify the fraction by dividing both terms in the numerator by 3.✓ Proved
5. \[ = 5 \]
   limit algebra simplifySubstitute x = 2/3 into the simplified expression. Perform the multiplication. Calculate the final value.✓ Proved