Properties of Limit at Infinity of Real Function/Combined Sum Rule

Theorem

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

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:

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

Proof

From Properties of Limit at Positive Infinity of Real Function: Multiple Rule, we have:

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

and:

$\ds \lim_{x \mathop \to \infty} \paren {\beta \map f x} = \beta L_2$

From Properties of Limit at Positive Infinity of Real Function: Sum Rule, we have:

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

$\blacksquare$