Limit of \( \displaystyle \frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3} \) as \( x \to \frac{3}{2} \)

Problem 1.140 · medium

1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3}\right) \]
   limitStarting with the limit of the function at x = 3/2.✓ Proved
2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{4 x^{2} - 2 x - 6}{2 x - 3}\right) \]
   algebra simplify factorExpand the squared term. Combine like terms in the numerator. Factor out a 2 from the numerator.✓ Proved
3. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{\left(x + 1\right) \left(4 x - 6\right)}{2 x - 3}\right) \]
   factorFactor the quadratic expression in the numerator.✓ Proved
4. \[ = \lim_{x \to \frac{3}{2}^+}\left(2 x + 2\right) \]
   cancel simplifyCancel the common factor (2*x - 3). Distribute the 2.✓ Proved
5. \[ = 5 \]
   substitution algebra limitSubstitute x = 3/2 into the expression. Multiply 2 by 3/2. Final evaluation of the limit.✓ Proved