The idea of a limit is key to understanding the fundamental concepts of calculus. There is a precise definition that formalizes this idea, and that is also sometimes useful (although the idea is more important).

What does it mean to increase without bound?

Here are two functions: Do they increase forever, or eventually reach a maximum? In fact, the red curve is $y = \log x \n,$ which increases forever, while the purple curve is $y = 3 \arctan 0.2x \n,$ which has a maximum of $y = 1.5\pi \approx 4.7 \n.$ There is really no way to know just from the graph, as both curves look very similar! So if you want to determine when a function has a maximum value, you have to look at the actual definition of the function. But how can you convince someone that a function increases forever, if you can’t just point to the graph and claim it’s obvious?

Let’s examine the idea of has no maximum. That means that, given some supposed maximum, the function exceeds that value at some point. So we might say

The function $f$ increases forever if for every number $M \n,$ there is some value $x$ such that $f(x) > M \n.$

Consider the function $f(x) = x + 2 \sin x \n,$ whose graph is shown below: That would satisfy our definition, since the graph will eventually rise high enough to cross any given horizontal line. But it wouldn’t be quite right to say it increases forever, since it also decreases sometimes. Perhaps a better term would be becomes arbitrarily large. The reason we use the word arbitrary is that someone can pick any value they would like, completely arbitrarily, and that number will not be a maximum for the function, no matter how large it was.