1. **State the problem:** Find the limit as $x$ approaches $+\infty$ of the function $\ln\left(\frac{x+1}{x}\right)$. 2. **Recall the limit and logarithm properties:** The natural logarithm function $\ln(x)$ is continuous and the limit of a logarithm is the logarithm of the limit if the limit inside exists and is positive. 3. **Simplify the argument inside the logarithm:** $$\frac{x+1}{x} = \frac{x}{x} + \frac{1}{x} = 1 + \frac{1}{x}$$ 4. **Rewrite the limit:** $$\lim_{x \to +\infty} \ln\left(1 + \frac{1}{x}\right)$$ 5. **Evaluate the inner limit:** As $x \to +\infty$, $\frac{1}{x} \to 0$, so $$\lim_{x \to +\infty} \left(1 + \frac{1}{x}\right) = 1 + 0 = 1$$ 6. **Apply continuity of $\ln$:** $$\lim_{x \to +\infty} \ln\left(1 + \frac{1}{x}\right) = \ln\left(\lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)\right) = \ln(1)$$ 7. **Calculate $\ln(1)$:** $$\ln(1) = 0$$ **Final answer:** $$\boxed{0}$$