Limit of \( \displaystyle \frac{2 x - 3}{\sqrt{2 x + 1} - 2} \) as \( x \to \frac{3}{2} \)

Problem 1.142 · medium

1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x - 3}{\sqrt{2 x + 1} - 2}\right) \]
   limit algebraIdentify the limit to evaluate. Multiply the numerator and denominator by the conjugate.✓ Proved
2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\sqrt{2 x + 1} + 2\right) \]
   algebra simplify cancelExpand the denominator using the difference of squares. Simplify the denominator. Cancel the common factor (2*x - 3).✓ Proved
3. \[ = 4 \]
   limit algebra simplify simplify simplifyEvaluate the limit by substituting x = 3/2. Simplify the expression inside the square root. Calculate the square root. Evaluate the square root. Final result.✓ Proved