1. **State the problem:** Find the limit of the function $$f(x) = \sqrt{x - 1} - \sqrt{x} \ln(x)$$ as $$x \to +\infty$$. 2. **Recall the behavior of components:** - As $$x \to +\infty$$, $$\sqrt{x - 1} \sim \sqrt{x}$$ because subtracting 1 becomes negligible. - The term $$\sqrt{x} \ln(x)$$ grows faster than $$\sqrt{x}$$ alone because $$\ln(x)$$ increases without bound, though slower than any power of $$x$$. 3. **Rewrite the function for large $$x$$:** $$f(x) \approx \sqrt{x} - \sqrt{x} \ln(x) = \sqrt{x}(1 - \ln(x))$$ 4. **Analyze the limit:** - Since $$\ln(x) \to +\infty$$ as $$x \to +\infty$$, the term $$1 - \ln(x) \to -\infty$$. - Multiplying by $$\sqrt{x}$$, which also tends to $$+\infty$$, the product $$\sqrt{x}(1 - \ln(x)) \to -\infty$$ because the negative logarithm dominates. 5. **Conclusion:** $$\lim_{x \to +\infty} f(x) = -\infty$$. This means the function decreases without bound as $$x$$ becomes very large.