Limit of \( \displaystyle \frac{- 6 x + 2 \left(2 x + 1\right)^{2} - 5}{2 x - 1} \) as \( x \to \frac{1}{2} \)

Problem 1.230 · hard

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{- 6 x + 2 \left(2 x + 1\right)^{2} - 5}{2 x - 1}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{8 x^{2} + 2 x - 3}{2 x - 1}\right) \]
   simplifyExpand and simplify the numerator.✓ Proved
3. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(8 x^{2} + 2 x - 3\right)}{\frac{d}{d x} \left(2 x - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule since the limit is of the form 0/0.✓ Proved
4. \[ = \lim_{x \to \frac{1}{2}^+}\left(8 x + 1\right) \]
   simplify simplifyCompute the derivatives. Simplify the resulting expression.✓ Proved
5. \[ = 5 \]
   limitEvaluate the limit at x = 1/2.✓ Proved