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

Problem 1.163 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + \left(2 x + 1\right)^{2} - 5}{2 x - 1}\right) \]
   limitEvaluate the limit of the given function at x = 1/2.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} + 6 x - 4}{2 x - 1}\right) \]
   simplify simplify algebraExpand the squared term in the numerator. Combine like terms in the numerator. Split the middle term to prepare for factoring.✓ Proved
3. \[ = \lim_{x \to \frac{1}{2}^+}\left(2 x + 4\right) \]
   factor cancelFactor the numerator. Cancel the common factor (2*x - 1).✓ Proved
4. \[ = 5 \]
   limit simplifySubstitute x = 1/2 into the simplified expression. Calculate the final value.✓ Proved