1. The problem is to find the limit $$\lim_{x \to 1} \log_{0.1} x$$. 2. Recall that the logarithm function $$\log_a x$$ is defined for $$a > 0$$ and $$a \neq 1$$, and it is continuous for $$x > 0$$. 3. Since $$0.1$$ is a positive number less than 1, the logarithm base is valid. 4. The limit of a continuous function at a point is the function value at that point, so $$\lim_{x \to 1} \log_{0.1} x = \log_{0.1} 1$$. 5. We know that $$\log_a 1 = 0$$ for any valid base $$a$$. 6. Therefore, $$\lim_{x \to 1} \log_{0.1} x = 0$$. Final answer: $$0$$