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

Problem 1.235 · medium

1. \[ \lim_{x \to 2^+}\left(\frac{- x + \left(x - 1\right)^{2} + 1}{\left(x - 1\right)^{2} - 1}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to 2^+}\left(\frac{x^{2} - 3 x + 2}{x^{2} - 2 x}\right) \]
   simplifyExpand the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to 2^+}\left(\frac{x - 1}{x}\right) \]
   factor cancelFactor both the numerator and the denominator. Cancel the common factor (x - 2).✓ Proved
4. \[ = \frac{1}{2} \]
   limitEvaluate the limit by substituting x = 2.✓ Proved