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

Problem 1.258 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{\sin^{2}{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}\right) \]
   limit algebraStart with the original limit expression. Rewrite the square of the fraction as the square of the sine term divided by the square of the denominator.✓ Proved
2. \[ = \left(\lim_{x \to 1^+}\left(\frac{\sin{\left(x - 1 \right)}}{x - 1}\right)\right)^{2} \]
   limit-lawApply the power rule for limits.✓ Proved
3. \[ = \left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{2} \]
   substitutionUse substitution u = x - 1, so as x approaches 1, u approaches 0.✓ Proved
4. \[ = 1 \]
   trig-limit simplifyApply the fundamental trigonometric limit lim_{u->0} sin(u)/u = 1. Evaluate the final numerical value.✓ Proved