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

Problem 1.419 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{1 - \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}\right) \]
   limitStart with the limit from the right.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x - 1 \right)}\right)}{\frac{d}{d x} \left(x - 1\right)^{2}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to 1^+}\left(\frac{\sin{\left(x - 1 \right)}}{2 x - 2}\right) \]
   simplifyCompute the derivatives.✓ Proved
4. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \sin{\left(x - 1 \right)}}{\frac{d}{d x} \left(2 x - 2\right)}\right) \]
   lhopitalApply L'Hopital's Rule again because the limit is of the form 0/0.✓ Proved
5. \[ = \lim_{x \to 1^+}\left(\frac{\cos{\left(x - 1 \right)}}{2}\right) \]
   simplifyCompute the derivatives.✓ Proved
6. \[ = \frac{1}{2} \]
   limitEvaluate the limit by plugging in x = 1.✓ Proved