Definition:Limit of Real Function/Limit at Infinity/Positive

Definition

Let $f$ be a real function defined on an open interval $\openint a \to$.

Let $L \in \R$.

$L$ is the limit of $f$ at infinity if and only if:

$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \size {\map f x - L} < \epsilon$

This is denoted as:

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

Increasing Without Bound

Suppose that:

$\forall M \in \R_{>0}: \exists N \in \R_{>0}: \forall x > N : \map f x > M$

for $M$ sufficiently large.

Then we write:

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

or

$\map f x \to +\infty$ as $x \to +\infty$

That is, $\map f x$ can be made arbitrarily large by making $x$ sufficiently large.

This is voiced:

$\map f x$ increases without bound as $x$ increases without bound.

or:

$\map f x$ tends to (plus) infinity as $x$ tends to (plus) infinity.

Decreasing Without Bound

Suppose that:

$\forall M \in \R_{<0}: \exists N \in \R_{>0}: x > N \implies \map f x < M$

for $M$ sufficiently large in magnitude.

Then we write:

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

or

$\map f x \to -\infty$ as $x \to +\infty$

That is, $-\map f x$ can be made arbitrarily large by making $x$ sufficiently large.

This is voiced:

$\map f x$ decreases without bound as $x$ increases without bound.

or:

$\map f x$ tends to minus infinity as $x$ tends to (plus) infinity.

Also see

• Definition:Limit at Minus Infinity

Sources

• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions
• 1970: N.G. de Bruijn: Asymptotic Methods in Analysis (3rd ed.) ... (previous) ... (next): $1.1$ What is asymptotics? $(1.1.3)$, $(1.1.4)$