1. **Problem Statement:** Define the following terms with examples: (a) ε - δ definition of a limit of a function μ(x) (b) Orthogonal vectors (c) The limit of a function f(x) at a point a (d) The conjugate of a complex number z 2. **Formulas and Definitions:** (a) The ε - δ definition of limit states: For every ε > 0, there exists a δ > 0 such that if $0 < |x - a| < \delta$, then $|\mu(x) - L| < \varepsilon$, where $L$ is the limit of $\mu(x)$ as $x \to a$. (b) Orthogonal vectors are vectors whose dot product is zero: For vectors $\mathbf{u}$ and $\mathbf{v}$, $\mathbf{u} \cdot \mathbf{v} = 0$. Example: $\mathbf{u} = (1,0)$ and $\mathbf{v} = (0,1)$. (c) The limit of $f(x)$ at $a$ is the value $L$ that $f(x)$ approaches as $x$ approaches $a$, denoted as $\lim_{x \to a} f(x) = L$. (d) The conjugate of a complex number $z = a + bi$ is $\overline{z} = a - bi$. 3. **Intermediate Work and Explanation:** (a) The ε - δ definition rigorously captures the idea of a limit by controlling how close $x$ must be to $a$ (within $\delta$) to ensure $\mu(x)$ is within $\varepsilon$ of $L$. (b) Orthogonal vectors have no projection on each other, meaning they are perpendicular in Euclidean space. (c) The limit describes the behavior of $f(x)$ near $a$, not necessarily at $a$. (d) The conjugate reflects the complex number across the real axis in the complex plane. **Final answers:** (a) $\forall \varepsilon > 0, \exists \delta > 0$ such that if $0 < |x - a| < \delta$, then $|\mu(x) - L| < \varepsilon$. (b) $\mathbf{u} \cdot \mathbf{v} = 0$; example: $(1,0)$ and $(0,1)$. (c) $\lim_{x \to a} f(x) = L$. (d) $\overline{z} = a - bi$ for $z = a + bi$.