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

Problem 1.556 · medium

1. \[ \lim_{x \to -1^+}\left(\frac{1 - \cos{\left(x + 1 \right)}}{\left(x + 1\right)^{2}}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{1 - \cos{\left(u \right)}}{u^{2}}\right) \]
   substitutionSubstitute u = x + 1, which implies u approaches 0 as x approaches -1.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \left(1 - \cos{\left(u \right)}\right)}{\frac{d}{d u} u^{2}}\right) \]
   lhopitalApply L'Hôpital's rule since the limit is in the indeterminate form 0/0.✓ Proved
4. \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{2 u}\right) \]
   simplify algebraCompute the derivatives of the numerator and denominator. Factor out the constant 1/2.✓ Proved
5. \[ = \lim_{u \to 0^+} \frac{1}{2} \]
   trig-limitUse the fundamental trigonometric limit sin(u)/u -> 1 as u approaches 0.✓ Proved
6. \[ = \frac{1}{2} \]
   limitEvaluate the final limit.✓ Proved