Category:Properties of Limit at Infinity of Real Function

This category contains pages concerning Properties of Limit at Infinity of Real Function:

Sum Rule

Let $a \in \R$.

Let $f, g : \hointr a \infty \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to \infty} \map f x = L_1$

and:

$\ds \lim_{x \mathop \to \infty} \map g x = L_2$

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:

$\ds \lim_{x \mathop \to \infty} \paren {\map f x + \map g x} = L_1 + L_2$

Multiple Rule

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

Let $f : \hointr a \infty \to \R$ be a real function such that:

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

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:

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

Combined Sum Rule

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

Let $f, g : \hointr a \infty \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to \infty} \map f x = L_1$

and:

$\ds \lim_{x \mathop \to \infty} \map g x = L_2$

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

Then:

$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x + \beta \map g x} = \alpha L_1 + \beta L_2$

Difference Rule

Let $a \in \R$.

Let $f, g : \hointr a \infty \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to \infty} \map f x = L_1$

and:

$\ds \lim_{x \mathop \to \infty} \map g x = L_2$

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

Then:

$\ds \lim_{x \mathop \to \infty} \paren {\map f x - \map g x} = L_1 - L_2$

Product Rule

Let $a \in \R$.

Let $f, g : \hointr a \infty \to \R$ be real functions such that:

$\ds \lim_{x \mathop \to \infty} \map f x = L_1$

and:

$\ds \lim_{x \mathop \to \infty} \map g x = L_2$

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

Then:

$\ds \lim_{x \mathop \to \infty} \paren {\map f x \map g x} = L_1 L_2$