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

Problem 1.29 · medium

1. \[ \lim_{x \to 3^+}\left(\frac{\sqrt{3} \sqrt{x} - 3}{3 x - 9}\right) \]
   limit factorStart with the limit of the function as x approaches 3 from the right. Factor the denominator to reveal the indeterminate form.✓ Proved
2. \[ = \lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \left(\sqrt{3} \sqrt{x} - 3\right)}{\frac{d}{d x} \left(3 x - 9\right)}\right) \]
   lhopitalApply L'Hopital's rule since the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to 3^+}\left(\frac{\sqrt{3}}{6 \sqrt{x}}\right) \]
   simplify algebraCompute the derivatives of the numerator and denominator. Simplify the fraction.✓ Proved
4. \[ = \frac{1}{6} \]
   limit simplifyEvaluate the limit by substituting x = 3. Simplify the resulting expression.✓ Proved