Properties of Limit at Infinity of Real Function/Difference Rule

Theorem

Let $a \in \R$.

Let $f, g : \hointr a \infty \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to \infty} \map f x = L_1$

and:

$\ds \lim_{x \mathop \to \infty} \map g x = L_2$

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

Then:

$\ds \lim_{x \mathop \to \infty} \paren {\map f x - \map g x} = L_1 - L_2$

Proof

This follows immediately by plugging $\alpha = 1$ and $\beta = -1$ into the combined sum rule.

$\blacksquare$