Properties of Limit at Infinity of Real Function/Sum 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:

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

with:

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

Since:

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

given $\epsilon > 0$, we can find $M_1 \ge 0$ such that:

$\ds \size {\map f x - L_1} < \frac \epsilon 2$ for $x \ge M_1$

Since:

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

we can find $M_2 \ge 0$ such that:

$\ds \size {\map f x - L_2} < \frac \epsilon 2$ for $x \ge M_2$.

Let:

$M = \max \set {M_1, M_2}$

Then, for $x \ge M$, we have:

$\ds \size {\paren {\map f x + \map g x} - \paren {L_1 + L_2} }$

$=$

$\ds \size {\paren {\map f x - L_1} + \paren {\map g x - L_2} }$

$\ds $

$\le$

$\ds \size {\map f x - L_1} + \size {\map g x - L_2}$

$\ds $

$<$

$\ds \frac \epsilon 2 + \frac \epsilon 2$

$\ds $

$=$

$\ds \epsilon$

Since $\epsilon$ was arbitrary, we have:

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

$\blacksquare$