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

Problem 1.125 · medium

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x + \left(3 x + 1\right)^{2} - 5}{3 x - 1}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 9 x - 4}{3 x - 1}\right) \]
   simplify simplifyExpand the squared term. Combine like terms in the numerator.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(9 x^{2} + 9 x - 4\right)}{\frac{d}{d x} \left(3 x - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
4. \[ = \lim_{x \to \frac{1}{3}^+}\left(6 x + 3\right) \]
   simplify simplifyCompute the derivatives. Simplify the resulting expression.✓ Proved
5. \[ = 5 \]
   limitEvaluate the limit by substituting x = 1/3.✓ Proved