1. **State the problem:** Find the limit as $x \to +\infty$ of the expression $$\frac{\sqrt{x} + \sqrt{x} + \sqrt{x}}{\sqrt{x} + 1}.$$\n\n2. **Simplify the numerator:** The numerator is $\sqrt{x} + \sqrt{x} + \sqrt{x} = 3\sqrt{x}$.\n\n3. **Rewrite the limit:** $$\lim_{x \to +\infty} \frac{3\sqrt{x}}{\sqrt{x} + 1}.$$\n\n4. **Divide numerator and denominator by $\sqrt{x}$ to simplify:** $$\lim_{x \to +\infty} \frac{3\sqrt{x}/\sqrt{x}}{(\sqrt{x} + 1)/\sqrt{x}} = \lim_{x \to +\infty} \frac{3}{1 + \frac{1}{\sqrt{x}}}.$$\n\n5. **Evaluate the limit:** As $x \to +\infty$, $\frac{1}{\sqrt{x}} \to 0$, so the expression becomes $$\frac{3}{1 + 0} = 3.$$\n\n**Final answer:** $$\boxed{3}.$$