Limit of \( \displaystyle \frac{\sqrt{\left(2 x + 1\right)^{2} + 9} - 3}{\left(2 x + 1\right)^{2}} \) as \( x \to - \frac{1}{2} \)

Problem 1.178 · hard

1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sqrt{\left(2 x + 1\right)^{2} + 9} - 3}{\left(2 x + 1\right)^{2}}\right) \]
   limit algebraStart with the original limit. Rationalize the numerator by multiplying by the conjugate.✓ Proved
2. \[ = \lim_{x \to - \frac{1}{2}^+} \frac{1}{\sqrt{\left(2 x + 1\right)^{2} + 9} + 3} \]
   algebra simplify cancelExpand the product in the numerator. Simplify the numerator. Cancel the common factor (2*x + 1)**2.✓ Proved
3. \[ = \lim_{x \to - \frac{1}{2}^+} \frac{1}{6} \]
   limitEvaluate the limit by substituting x = -1/2.✓ Proved
4. \[ = \frac{1}{6} \]
   algebra simplify simplifySimplify the expression inside the square root. Calculate the square root. Final result.✓ Proved