Combination Theorem for Limits of Functions

Real Functions

Let $\R$ denote the real numbers.

Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:

$\ds \lim_{x \mathop \to c} \map f x = l$

$\ds \lim_{x \mathop \to c} \map g x = m$

Let $\lambda, \mu \in \R$ be arbitrary real numbers.

Then the following results hold:

Sum Rule

$\ds \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$

Multiple Rule

$\ds \lim_{x \mathop \to c} \lambda \map f x = \lambda l$

Combined Sum Rule

$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$

Product Rule

$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$

Quotient Rule

$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$

provided that $m \ne 0$.

Complex Functions

Let $\C$ denote the complex numbers.

Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:

$\ds \lim_{z \mathop \to c} \map f z = l$

$\ds \lim_{z \mathop \to c} \map g z = m$

Let $\lambda, \mu \in \C$ be arbitrary complex numbers.

Then the following results hold:

Sum Rule

$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$

Multiple Rule

$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$

Combined Sum Rule

$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$

Product Rule

$\ds \lim_{z \mathop \to c} \ \paren {\map f z \map g z} = l m$

Quotient Rule

$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$

provided that $m \ne 0$.