1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{1 - \cos x}{x^2}$$ 2. **Recall the formula and important rules:** We use the Taylor series expansion for cosine near 0: $$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots$$ This helps us approximate the numerator for small $x$. 3. **Substitute the expansion into the limit expression:** $$\frac{1 - \cos x}{x^2} = \frac{1 - \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)}{x^2} = \frac{\frac{x^2}{2} - \frac{x^4}{24} + \cdots}{x^2}$$ 4. **Simplify the fraction by dividing numerator and denominator by $x^2$:** $$= \frac{\cancel{x^2} \left(\frac{1}{2} - \frac{x^2}{24} + \cdots \right)}{\cancel{x^2}} = \frac{1}{2} - \frac{x^2}{24} + \cdots$$ 5. **Evaluate the limit as $x \to 0$:** Since higher order terms vanish, $$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$$ **Final answer:** $$\boxed{\frac{1}{2}}$$