Properties of Limit at Minus Infinity of Real Function/Difference Rule

Theorem

Let $a, \alpha, \beta \in \R$.

Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist

where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.

Then:

$\ds \lim_{x \mathop \to -\infty} \paren {\map f x - \map g x}$ exists

with:

$\ds \lim_{x \mathop \to -\infty} \paren {\map f x - \map g x} = \lim_{x \mathop \to \infty} \map f x - \lim_{x \mathop \to \infty} \map g x$

Proof

This is the combined sum rule with $\alpha = 1$, $\beta = -1$.

$\blacksquare$