Definition:Infinite Limit at Infinity/Increasing Without Bound

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Definition

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

Suppose that:

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

for $M$ sufficiently large.

Then we write:

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

or

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


That is, $f\left({x}\right)$ can be made arbitrarily large by making $x$ sufficiently large.

This is voiced:

$f\left({x}\right)$ increases without bound as $x$ increases without bound.


Similarly, suppose:

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

for $M$ sufficiently large.

Then we write:

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

or

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

This is voiced:

$f\left({x}\right)$ increases without bound as $x$ decreases without bound.


That is, you can make $f\left({x}\right)$ arbitrarily large by making $x$ sufficiently large in magnitude.


Intuition

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

If I tell you:

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

then 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 $N$.