Combination Theorem for Limits of Mappings/Metric Space

Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $\R$ denote the real numbers.

Let $f: A \to \R$ and $g: A \to \R$ be real-valued functions defined on $A$, except possibly at the point $a \in A$.

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

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

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

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

Then the following results hold:

Sum Rule

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

Multiple Rule

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

Product Rule

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

Quotient Rule

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

provided that $m \ne 0$.