Properties of Limit at Minus Infinity of Real Function/Combined Sum 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:

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

with:

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

Proof

From Properties of Limit at Minus Infinity of Real Function: Multiple Rule:

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

with:

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

and:

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

with:

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

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

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

with:

\(\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x + \beta \map g x}\)

\(=\)

\(\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} + \lim_{x \mathop \to -\infty} \paren {\beta \map g x}\)

\(\ds \)

\(=\)

\(\ds \alpha \lim_{x \mathop \to -\infty} \map f x + \beta \lim_{x \mathop \to -\infty} \map g x\)

$\blacksquare$