Combination Theorem for Limits of Functions/Real

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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 $m \ne 0$.

Also see