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

Theorem

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

Let $f : \hointl {-\infty} a \to \R$ be a real function such that:

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

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

Then:

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

with:

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

Proof

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

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

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

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

with:

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

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

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

$\blacksquare$