Limit of \( \displaystyle \frac{\sqrt{3 x + 5} - 2}{3 x + 1} \) as \( x \to - \frac{1}{3} \)

Problem 1.35 · medium

1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 5} - 2}{3 x + 1}\right) \]
   limit algebraSet up the limit from the right. Multiply the numerator and denominator by the conjugate.✓ Proved
2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 5} + 2} \]
   simplify algebra cancelSimplify the numerator using the difference of squares. Group the terms to prepare for cancellation. Cancel the common factor (3*x + 1).✓ Proved
3. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{4} \]
   limit simplify simplify simplifyEvaluate the limit by substituting x = -1/3. Simplify the expression inside the square root. Calculate the square root. Add the numbers in the denominator.✓ Proved
4. \[ = \frac{1}{4} \]
   simplifyFinal result.✓ Proved