Combination Theorem for Limits of Functions/Real

Theorem
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:

Also see

 * Combination Theorem for Continuous Functions