1. **State the problem:** Find the limit $$\lim_{x\to 2} \frac{\ln x - \ln 2}{x - 2}$$. 2. **Recall the formula:** This limit resembles the definition of the derivative of the function $f(x) = \ln x$ at $x=2$: $$\lim_{x\to a} \frac{f(x) - f(a)}{x - a} = f'(a)$$. 3. **Derivative of $\ln x$:** The derivative is $$f'(x) = \frac{1}{x}$$. 4. **Evaluate the derivative at $x=2$:** $$f'(2) = \frac{1}{2}$$. 5. **Conclusion:** Therefore, $$\lim_{x\to 2} \frac{\ln x - \ln 2}{x - 2} = \frac{1}{2}$$. This limit represents the slope of the tangent line to $y=\ln x$ at $x=2$.