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

Problem 1.103 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{1 - \cos{\left(x - 1 \right)}}{x - 1}\right) \]
   limitStart with the limit of the original function.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x - 1 \right)}\right)}{\frac{d}{d x} \left(x - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 1^+} \sin{\left(x - 1 \right)} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression.✓ Proved
4. \[ = 0 \]
   limit simplifyEvaluate the limit by substituting x = 1. Final result.✓ Proved