1. **Problem:** Calculate the limit $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2}$$ **Step 1:** Recall the important limit definition of the number $e$: $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$ **Step 2:** We can rewrite the expression as: $$\left(1 + \frac{1}{x}\right)^{x+2} = \left(1 + \frac{1}{x}\right)^x \cdot \left(1 + \frac{1}{x}\right)^2$$ **Step 3:** As $x \to \infty$, $\left(1 + \frac{1}{x}\right)^x \to e$ and $\left(1 + \frac{1}{x}\right)^2 \to 1^2 = 1$ **Step 4:** Therefore, $$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+2} = e \times 1 = e$$ **Final answer:** $e$ 2. **Problem:** Calculate the limit $$\lim_{x \to \infty} \left(\frac{x+3}{x-1}\right)^{x+2}$$ **Step 1:** Simplify the base: $$\frac{x+3}{x-1} = \frac{x-1+4}{x-1} = 1 + \frac{4}{x-1}$$ **Step 2:** Rewrite the limit as: $$\lim_{x \to \infty} \left(1 + \frac{4}{x-1}\right)^{x+2}$$ **Step 3:** For large $x$, $x+2 \approx x$ and $x-1 \approx x$, so approximate as: $$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^x$$ **Step 4:** Using the limit definition of $e$: $$\lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^x = e^k$$ **Step 5:** Here, $k=4$, so the limit is: $$e^4$$ **Final answer:** $e^4$ 3. **Problem:** Calculate the limit $$\lim_{x \to \pi} (\cos 2x)^{x^2}$$ **Step 1:** Evaluate the base at $x=\pi$: $$\cos 2\pi = \cos 0 = 1$$ **Step 2:** Evaluate the exponent at $x=\pi$: $$\pi^2$$ **Step 3:** Since the base approaches 1 and the exponent approaches $\pi^2$, the limit is: $$1^{\pi^2} = 1$$ **Final answer:** $1$ 4. **Problem:** Calculate the limit $$\lim_{x \to 0} \frac{\tan 3x}{\sin 3x}$$ **Step 1:** Recall the small angle approximations: $$\tan \theta \approx \theta, \quad \sin \theta \approx \theta \quad \text{as } \theta \to 0$$ **Step 2:** Substitute $\theta = 3x$: $$\frac{\tan 3x}{\sin 3x} \approx \frac{3x}{3x} = 1$$ **Step 3:** More rigorously, use the limit: $$\lim_{x \to 0} \frac{\tan 3x}{\sin 3x} = \lim_{x \to 0} \frac{\tan 3x / 3x}{\sin 3x / 3x} = \frac{1}{1} = 1$$ **Final answer:** $1$