1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{\sqrt{1+x} - 2}{x - 3}$$. 2. **Recall the formula and rules:** This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ which resembles the definition of the derivative of $$f(x) = \sqrt{1+x}$$ at $$x = 3$$. 3. **Evaluate the function at $$x=3$$:** $$f(3) = \sqrt{1+3} = \sqrt{4} = 2$$. 4. **Rewrite the limit as a derivative:** $$\lim_{x \to 3} \frac{\sqrt{1+x} - 2}{x - 3} = f'(3)$$. 5. **Find the derivative of $$f(x) = \sqrt{1+x}$$:** $$f'(x) = \frac{1}{2\sqrt{1+x}}$$. 6. **Evaluate the derivative at $$x=3$$:** $$f'(3) = \frac{1}{2\sqrt{1+3}} = \frac{1}{2\times 2} = \frac{1}{4}$$. 7. **Final answer:** $$\boxed{\frac{1}{4}}$$