Limits

Let $S \subseteq \mathbb{C}$, let $f: S \to \mathbb{C}$ be a function, and let $z_0$ be a limit point of $S$, so every deleted neighborhood of $z_0$ contains at least 1 point of $S$. For $\omega_0 \in \mathbb{C}$,

iff, for each $\epsilon > 0$, there exists $\delta > 0$ such that, for each $z \in S$,

$f(z)$ can be made arbitrarily close to $\omega_0$ by choosing $z$ sufficiently close to, but distinct from, $z_0$.

Uniqueness of Limits

If a limit of $f: S \to \mathbb{C}$ exists at a limit point of $S$, the limit is unique.

Limits via Real and Imaginary Parts

Let $f(z) = u(x,y)+iv(x,y)$, $z_0 = x_0+iy_0$, $\omega_0 = u_0+iv_0$. Then

iff

Limit Laws

Suppose $\displaystyle\lim_{z \to z_0} f(z) = \omega_0$ and $\displaystyle\lim_{z \to z_0} g(z) = \Omega_0$. Then:

• $\displaystyle\lim_{z \to z_0} (f(z)+g(z)) = \omega_0+\Omega_0$
• $\displaystyle\lim_{z \to z_0} (f(z)g(z)) = \omega_0\Omega_0$
• $\displaystyle\lim_{z \to z_0} \frac{f(z)}{g(z)} = \frac{\omega_0}{\Omega_0}$, provided $\Omega_0 \ne 0$

Limit of a Composition

Let $f: B \to \mathbb{C}$ and $g: A \to B$ be functions. Suppose

and $g(\omega) \ne z_0$ for all $\omega$ sufficiently close to $\omega_0$, $\omega \ne \omega_0$. Then

Limits Involving Infinity

Let $z_0, \omega_0 \in \mathbb{C}$.

• $\displaystyle\lim_{z \to z_0} f(z) = \infty$ iff $\displaystyle\lim_{z \to z_0} \frac{1}{f(z)} = 0$, where $f(z) \ne 0$ for all $z$ sufficiently close to $z_0$ with $z \ne z_0$
• $\displaystyle\lim_{z \to \infty} f(z) = \omega_0$ iff $\displaystyle\lim_{z \to 0} f\left(\frac{1}{z}\right) = \omega_0$
• $\displaystyle\lim_{z \to \infty} f(z) = \infty$ iff $\displaystyle\lim_{z \to 0} \frac{1}{f(1/z)} = 0$