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

Problem 1.229 · medium

1. \[ \lim_{x \to -1^+}\left(\frac{\sin^{2}{\left(x + 1 \right)}}{\left(x + 1\right)^{2}}\right) \]
   limit algebraIdentify the limit to evaluate. Rewrite the expression as a single squared term.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(u \right)}}{u^{2}}\right) \]
   substitutionLet u = x + 1, so as x approaches -1, 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. \[ = \lim_{u \to 0^+} 1 \]
   trig-limitUse the fundamental trigonometric limit sin(u)/u -> 1.✓ Proved
5. \[ = 1 \]
   simplifyThe limit evaluates to 1.✓ Proved