1. **Stating the problem:** We want to understand why $$\frac{\cos x - 1}{x} = 1$$ as $$x$$ approaches 0. 2. **Recall the limit definition and Taylor series:** The limit $$\lim_{x \to 0} \frac{\cos x - 1}{x}$$ is a classic limit in calculus. To analyze it, we use the Taylor series expansion of $$\cos x$$ around 0: $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$ 3. **Substitute the series into the expression:** $$\frac{\cos x - 1}{x} = \frac{\left(1 - \frac{x^2}{2} + \cdots\right) - 1}{x} = \frac{-\frac{x^2}{2} + \cdots}{x}$$ 4. **Simplify the fraction:** $$= \frac{-\frac{x^2}{2} + \cdots}{x} = -\frac{x}{2} + \cdots$$ 5. **Evaluate the limit:** As $$x \to 0$$, the term $$-\frac{x}{2}$$ approaches 0, and higher order terms vanish even faster. Therefore, $$\lim_{x \to 0} \frac{\cos x - 1}{x} = 0$$ 6. **Conclusion:** The limit is actually 0, not 1. So the statement $$\frac{\cos x - 1}{x} = 1$$ as $$x \to 0$$ is incorrect. If you meant a different expression or limit, please clarify.