Limit Ln 6116D8

1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{\ln x}{x-1}$$. 2. **Recall the formula and important rule:** This is a classic limit that can be evaluated using L'Hôpital's Rule because direct substitution gives an indeterminate form $\frac{0}{0}$. 3. **Apply L'Hôpital's Rule:** Differentiate numerator and denominator separately: $$\text{Numerator derivative} = \frac{d}{dx}(\ln x) = \frac{1}{x}$$ $$\text{Denominator derivative} = \frac{d}{dx}(x-1) = 1$$ 4. **Evaluate the new limit:** $$\lim_{x \to 1} \frac{1/x}{1} = \lim_{x \to 1} \frac{1}{x} = \frac{1}{1} = 1$$ 5. **Conclusion:** The limit is $$1$$. This limit is also the definition of the derivative of $\ln x$ at $x=1$.