Limit of \( \displaystyle \frac{1 - \cos{\left(2 x \right)}}{4 x^{2}} \) as \( x \to 0 \)

Problem 1.378 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(2 x \right)}}{4 x^{2}}\right) \]
   limitStart with the limit of the function as x approaches 0 from the right.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(2 x \right)}\right)}{\frac{d}{d x} 4 x^{2}}\right) \]
   lhopitalApply L'Hôpital's Rule because the limit is an indeterminate form of type 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{4 x}\right) \]
   simplify simplify rewriteCompute the derivatives of the numerator and denominator. Simplify the fraction by dividing both terms by 2. Rewrite the expression to isolate the fundamental trigonometric limit.✓ Proved
4. \[ = \frac{\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{2 x}\right)}{2} \]
   limit-lawApply the limit law for the product of a constant and a function.✓ Proved
5. \[ = \frac{1}{2} \]
   trig-limit simplifyUse the standard limit result that sin(u)/u approaches 1 as u approaches 0. Final evaluation.✓ Proved