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

Problem 1.248 · medium

1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sin^{2}{\left(3 x + 1 \right)}}{\left(3 x + 1\right)^{2}}\right) \]
   limit algebraStart with the original limit. Rewrite the expression using the power rule for quotients.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(u \right)}}{u^{2}}\right) \]
   substitutionLet u = 3*x + 1. As x approaches -1/3, u approaches 0.✓ Proved
3. \[ = \left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{2} \]
   limit-lawApply the power rule for limits.✓ Proved
4. \[ = 1 \]
   limit simplifyEvaluate the fundamental trigonometric limit. Final result.✓ Proved