Properties of Limit at Minus Infinity of Real Function

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Theorem

Relation with Limit at $+\infty$

Let $a \in \R$.

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


Then:

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

and in this case:

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

where:

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


Sum Rule

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:

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


Multiple Rule

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:

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


Combined Sum Rule

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:

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


Difference Rule

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:

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

with:

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


Product Rule

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:

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