Limit of \( \displaystyle \frac{3 x - 4}{\sqrt{3} \sqrt{x} - 2} \) as \( x \to \frac{4}{3} \)

Problem 1.301 · medium

1. \[ \lim_{x \to \frac{4}{3}^+}\left(\frac{3 x - 4}{\sqrt{3} \sqrt{x} - 2}\right) \]
   limitIdentify the limit and the indeterminate form 0/0.✓ Proved
2. \[ = \lim_{x \to \frac{4}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x - 4\right)}{\frac{d}{d x} \left(\sqrt{3} \sqrt{x} - 2\right)}\right) \]
   lhopitalApply L'Hôpital's Rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to \frac{4}{3}^+}\left(2 \sqrt{3} \sqrt{x}\right) \]
   simplify algebra algebra simplifyCompute the derivatives of the numerator and denominator. Simplify the complex fraction. Multiply the numerator by the reciprocal of the denominator. Rationalize the expression by multiplying numerator and denominator by sqrt(3).✓ Proved
4. \[ = 4 \]
   limitEvaluate the limit by substituting x = 4/3.✓ Proved