Definition:Infinite Limit at Infinity

Definition

Let $f$ be a real function defined on an open interval $\left({a .. +\infty}\right)$.

To say that:

$\displaystyle \lim_{x \to +\infty} f\left({x}\right) = +\infty$

is to say that:

$\forall M \in \R_{>0}: \exists N \in \R_{>0}: x > N \implies f \left({x}\right) > M$

That is, for every real positive $M$ there exists a real positive $N$ such that every real number in the domain of $f$ larger than of $N$ has an image larger than $M$.

Similarly, to say that:

$\displaystyle \lim_{x \to +\infty} f\left({x}\right) = -\infty$

means that:

$\forall M \in \R_{<0}: \exists N \in \R_{>0}: x > N \implies f \left({x}\right) < M$

Suppose that $f$ is defined on an open interval $\left({-\infty .. a}\right)$.

The statement:

$\displaystyle \lim_{x \to -\infty} f\left({x}\right) = +\infty$

asserts that:

$\forall M \in \R_{>0}: \exists N \in \R_{<0}: x < N \implies f \left({x}\right) > M$

Lastly:

$\displaystyle \lim_{x \to -\infty} f\left({x}\right) = -\infty$

is to say that:

$\forall M \in \R_{<0}: \exists N \in \R_{<0}: x < N \implies f \left({x}\right) < M$

$M$ and $N$ have the connotation of being very large in magnitude.

Note on terminology

In the above cases $f$ is said to increase or decrease without bound.

Intuition

You want to get very high on the $f\left({x}\right)$ axis. This degree of "highness" is the positive real number $M$.

I tell you:

$f\left({x}\right) \to +\infty$ as $x \to +\infty$

I am making you a promise. I guarantee you that there is a point on the $x$ axis that will satisfy your request. This value on the $x$ axis is the positive real number $N$.

(Now ask me for a larger $M$. I'll be here all day.)

Also see

• Limit of a function
• Infinite limit
• Limit at infinity
• Divergent to Infinity

Sources

• Roland E. Larson,  Robert P. Hostetler and Bruce H. Edwards: Calculus: 8th Edition (2005): $\S 3.5$