Category:Properties of Limit at Minus Infinity of Real Function
This category contains pages concerning Properties of Limit at Minus Infinity of Real Function:
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:
- $\ds \lim_{x \mathop \to -\infty} \paren {\map f x \map g x} = \paren {\lim_{x \mathop \to \infty} \map f x} \paren {\lim_{x \mathop \to \infty} \map g x}$
Pages in category "Properties of Limit at Minus Infinity of Real Function"
The following 7 pages are in this category, out of 7 total.
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- Properties of Limit at Minus Infinity of Real Function
- Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule
- Properties of Limit at Minus Infinity of Real Function/Difference Rule
- Properties of Limit at Minus Infinity of Real Function/Multiple Rule
- Properties of Limit at Minus Infinity of Real Function/Product Rule
- Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity
- Properties of Limit at Minus Infinity of Real Function/Sum Rule