1. State the problem: Find the limit $$\lim_{x\to 0} \frac{\ln(x+h) - \ln h}{x}$$ where $h>0$. 2. Recognize the expression as a difference quotient for the derivative of $\ln(x+h)$ at $x=0$. 3. Recall that the derivative of $\ln t$ with respect to $t$ is $\frac{1}{t}$. 4. Apply the definition of the derivative: $$\lim_{x\to 0} \frac{\ln(x+h) - \ln h}{x} = \left. \frac{d}{dx} \ln(x+h) \right|_{x=0} = \frac{1}{h}.$$ 5. Final answer: $$\boxed{\frac{1}{h}}$$.