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

Problem 1.96 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{- 3 x + 2 \left(x + 1\right)^{2} - 5}{x - 1}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{2 x^{2} + x - 3}{x - 1}\right) \]
   algebra algebra simplify rewriteExpand the squared term. Distribute the 2. Combine like terms in the numerator. Rewrite the numerator to reveal a factor of (x-1).✓ Proved
3. \[ = \lim_{x \to 1^+}\left(\frac{2 x \left(x - 1\right) + 3 x - 3}{x - 1}\right) \]
   factorFactor out (x - 1) from the numerator.✓ Proved
4. \[ = \lim_{x \to 1^+}\left(2 x + 3\right) \]
   simplifyCancel the common factor (x - 1).✓ Proved
5. \[ = 5 \]
   limitEvaluate the limit by substituting x = 1.✓ Proved