Properties of Limit at Minus Infinity of Real Function/Sum Rule
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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:
- $\ds \lim_{x \mathop \to -\infty} \paren {\map f x + \map g x} = \lim_{x \mathop \to \infty} \map f x + \lim_{x \mathop \to \infty} \map g x$
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: Sum Rule, we then have:
- $\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} + \map g {-x} }$ exists
with:
- $\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} + \map g {-x} } = \lim_{x \mathop \to \infty} \map f {-x} + \lim_{x \mathop \to \infty} \map g {-x}$
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, this gives:
- $\ds \lim_{x \mathop \to -\infty} \paren {\map f x + \map g x} = \lim_{x \mathop \to \infty} \map f x + \lim_{x \mathop \to \infty} \map g x$
$\blacksquare$