Limit of \( \displaystyle \frac{\sqrt{2} \sqrt{x} - 3}{2 x - 9} \) as \( x \to \frac{9}{2} \)

Problem 1.127 · medium

1. \[ \lim_{x \to \frac{9}{2}^+}\left(\frac{\sqrt{2} \sqrt{x} - 3}{2 x - 9}\right) \]
   limit limit algebra rewriteDefine the limit with the specified direction. Remove the direction specification for standard evaluation. Rewrite the numerator using the property sqrt(a)*sqrt(b) = sqrt(a*b). Multiply numerator and denominator by the conjugate.✓ Proved
2. \[ = \lim_{x \to \frac{9}{2}^+} \frac{1}{\sqrt{2} \sqrt{x} + 3} \]
   algebra simplifySimplify the numerator using the difference of squares. Cancel the common factor (2*x - 9).✓ Proved
3. \[ = \frac{1}{6} \]
   limit simplify simplifyEvaluate the limit by substituting x = 9/2. Simplify the square root. Final calculation.✓ Proved