1. **State the problem:** Evaluate the limit $$\lim_{w \to \infty} \left(\ln w - \sqrt{w}\right)$$ and the equivalent form $$\lim_{w \to \infty} \sqrt{w} \left(\frac{\ln w}{\sqrt{w}} - 1\right).\n 2. **Recall important rules:** - As $w \to \infty$, $\ln w$ grows slowly compared to any power of $w$. - $\sqrt{w} = w^{1/2}$ grows faster than $\ln w$. 3. **Analyze the original limit:** $$\lim_{w \to \infty} (\ln w - \sqrt{w}) = \lim_{w \to \infty} \left(-\sqrt{w} + \ln w\right).$$ Since $\sqrt{w}$ dominates $\ln w$, the term $-\sqrt{w}$ tends to $-\infty$ faster than $\ln w$ grows. 4. **Conclusion for the first limit:** $$\lim_{w \to \infty} (\ln w - \sqrt{w}) = -\infty.$$ 5. **Rewrite the limit as given:** $$\lim_{w \to \infty} \sqrt{w} \left(\frac{\ln w}{\sqrt{w}} - 1\right) = \lim_{w \to \infty} \left(\ln w - \sqrt{w}\right),$$ which is the same expression as before. 6. **Evaluate the rewritten limit:** Since it is the same as the original, the limit is also $$-\infty.$$