Limit of \( \displaystyle \frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1} \) as \( x \to 0 \)

Problem 1.41 · hard

1. \[ \lim_{x \to 0^+}\left(\frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1}\right) \]
   limitStart with the limit expression.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{4 x^{2} + 2 x}{4 x^{2} + 4 x}\right) \]
   algebra simplify simplifyExpand the squared terms. Combine like terms in the numerator and denominator. Combine like terms in the numerator.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{4 x + 2}{4 x + 4}\right) \]
   factor cancelFactor out x from both numerator and denominator. Cancel the common factor x.✓ Proved
4. \[ = \frac{1}{2} \]
   limitEvaluate the limit by substituting x = 0.✓ Proved