1. Problem: Find the limit $$\lim_{x \to 0} \frac{\cos x}{x+1}$$. 2. Formula and rules: The limit of a quotient is the quotient of the limits if both limits exist. Also, $$\lim_{x \to 0} \cos x = 1$$ and $$\lim_{x \to 0} (x+1) = 1$$. 3. Intermediate work: $$\lim_{x \to 0} \frac{\cos x}{x+1} = \frac{\lim_{x \to 0} \cos x}{\lim_{x \to 0} (x+1)} = \frac{1}{1}$$ 4. Explanation: As $$x$$ approaches 0, $$\cos x$$ approaches 1 and $$x+1$$ approaches 1, so the fraction approaches $$\frac{1}{1}$$. 5. Final answer: $$\boxed{1}$$