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

Problem 1.358 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{3 x - 3}{\sqrt{3 x + 1} - 2}\right) \]
   limit factor rationalizeStart with the limit of the function as x approaches 1 from the right. Factor out the constant 3 from the numerator. Multiply the numerator and denominator by the conjugate of the denominator.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\sqrt{3 x + 1} + 2\right) \]
   algebra simplify factor cancelExpand the denominator using the difference of squares. Simplify the expression in the denominator. Factor out the 3 from the denominator. Cancel the common factor (x - 1) from the numerator and denominator.✓ Proved
3. \[ = 4 \]
   limitEvaluate the limit by substituting x = 1.✓ Proved