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

Theorem

Let $a \in \R$.

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:

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

Proof

From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have:

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

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

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

with:

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

From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, this gives:

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

$\blacksquare$