1. **State the problem:** Find the limit $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2}$$. 2. **Recall the important formula:** The expression $$\left(1 + \frac{1}{x}\right)^x$$ approaches the number $$e$$ as $$x$$ approaches infinity, where $$e \approx 2.71828$$. 3. **Rewrite the limit:** $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2} = \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \cdot \left(1 + \frac{1}{x}\right)^2$$ 4. **Evaluate each part:** - As $$x \to \infty$$, $$\left(1 + \frac{1}{x}\right)^x \to e$$. - Also, $$\left(1 + \frac{1}{x}\right)^2 \to 1^2 = 1$$. 5. **Combine the results:** $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2} = e \times 1 = e$$. 6. **Final answer:** $$\boxed{e}$$