1. Prove in the sense of distributions
   $$\lim_{\varepsilon \to 0} \frac{\varepsilon}{\pi (x^2 + \varepsilon^2)} = \delta_0.$$

2. For any $\varepsilon > 0$, $\frac{1}{x+i\varepsilon}$ is a locally integrable function on $\mathbb{R}$. As a distribution on $\mathbb{R}$, prove
   $$\lim_{\varepsilon \to 0^+} \frac{1}{x+i\varepsilon} \overset{\mathcal{D}'}{=} \text{vp} \frac{1}{x} - i\pi \delta_0.$$
   The limit distribution can also be written as $\frac{1}{x+i0}$.

3. As a distribution on $\mathbb{R}$, $u_n(x) = x^{-1} \sin(nx)$, compute the limit in the distribution space $\mathcal{D}'(\mathbb{R})$ of $\lim_{n \to \infty} u_n$.

4. We denote the surface area of the unit sphere $|\mathbf S^{n-1}|$ in $\mathbb{R}^n$ by $\mathbf S^{n-1}$. Let
   $$E = \frac{|x|^{2-n}}{(2-n)|\mathbf S^{n-1}|}.$$
   Prove in the sense of distributions that
   $$\Delta E \overset{\mathcal{D}'}{=} \delta_0.$$

5. Assume $\Omega \subset \mathbb{R}^3$ is an open set and $u \in \mathcal{D}'(\Omega)$, satisfying
   $$\Delta u = f,$$
   where $f \in C^\infty(\Omega)$ is a smooth function. Hence, $u \in C^\infty(\Omega)$.

6. Suppose $f \in \mathcal{E}'(\mathbb{R}^3)$ (a distribution with compact support). Prove the existence of a constant $C_1 > 0$ (which may depend on the distribution $f$), such that, as $|x| \to \infty$,
   $$|E * f(x)| \leq \frac{C}{|x|}.$$

7. Assume $f \in \mathcal{E}'(\mathbb{R}^3)$. Prove that $u = E * f$ is a solution to the potential equation
   $$\Delta u = f.$$
   Furthermore, a constant $C_2 > 0$ exists such that, as $|x| \to \infty$,
   $$\left| u(x) - \frac{\langle f, 1 \rangle}{4\pi |x|} \right| \leq \frac{C_2}{|x|^2}.$$
   Depending on the physical context, $\langle f, 1 \rangle$ represents the total mass or total charge of $u$.

8. Suppose $u \in C^\infty(\mathbb{R}^3)$ is a real-valued function, and define the sign function $\operatorname{sign}(u)$ as
   $$\operatorname{sign}(u)(x) = \begin{cases} 1, & u(x) > 0; \\ 0, & u(x) = 0; \\ -1, & u(x) < 0. \end{cases}$$
   Prove, in the sense of distributions that
   $$\Delta |u| \overset{\mathcal{D}'}{\geq} \Delta u \cdot \operatorname{sign}(u),$$
   i.e., for any non-negative real-valued function $\varphi \in \mathcal{D}(\mathbb{R}^3)$,
   $$\langle \Delta |u|, \varphi \rangle \geq \langle \Delta u \cdot \operatorname{sign}(u), \varphi \rangle,$$
   Hint: First approximate $u$ in the sense of distributions with $\sqrt{u^2 + \varepsilon^2}$.