Combination Theorem for Limits of Functions/Real

Theorem
Let $X$ be one of the standard number fields $\Q, \R, \C$.

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

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


 * $\displaystyle \lim_{x \to c} \ f \left({x}\right) = l$
 * $\displaystyle \lim_{x \to c} \ g \left({x}\right) = m$

Let $\lambda, \mu \in X$ be arbitrary numbers in $X$.

Then the following results hold:

Also see

 * Combination Theorem for Continuous Functions