Limit of \( \displaystyle \frac{1 - \cos{\left(x \right)}}{x^{2}} \) as \( x \to 0 \)

Problem 1.305 · easy

1. \[ \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{x^{2}}\right) \]
   limitIdentify the limit to be evaluated.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}{\frac{d}{d x} x^{2}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{2 x}\right) \]
   simplifyDifferentiate the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{2} \]
   limit-lawSeparate the constant factor from the limit.✓ Proved
5. \[ = \frac{\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)}{2} \]
   simplifyEvaluate the constant factor.✓ Proved
6. \[ = \frac{\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} x}\right)}{2} \]
   lhopitalApply L'Hopital's rule to the remaining indeterminate form.✓ Proved
7. \[ = \frac{\lim_{x \to 0^+} \cos{\left(x \right)}}{2} \]
   simplify limit-lawDifferentiate the numerator and denominator. Apply the limit to the function part.✓ Proved
8. \[ = \frac{1}{2} \]
   limit simplifyEvaluate the limit of cos(x) at 0. Final result.✓ Proved