Infinite limit at infinity

The limit of a function is one of the most important ideas in mathematical analysis. It describes how a function behaves near a particular point, which lets us define, for example, the continuity of a function. Limits help us understand how a function behaves even at places where it is not defined at all.

Infinite limit at infinity

This combines the previous limits – we look for the limit of a function as x tends to plus or minus infinity, and the limit itself also turns out to be plus or minus infinity.

We say that ∞ is the limit of the function at the point ∞ if

$$(\mathrm{\forall }K\in \mathbb{R})(\mathrm{\exists }A\in \mathbb{R})(\mathrm{\forall }x\in \mathrm{dom}(f))(x>A\Rightarrow f(x)>K)$$

and at the point −∞ if

$$(\mathrm{\forall }K\in \mathbb{R})(\mathrm{\exists }A\in \mathbb{R})(\mathrm{\forall }x\in \mathrm{dom}(f))(x<A\Rightarrow f(x)>K).$$

It is similar for x tending to −∞. The definition combines the previous ideas. It tells us that if we are looking for a limit at infinity and that limit is to be infinity again, then for every bound K on the y-axis we can find a bound A on the x-axis such that all the function values f(x) for x > A, that is, beyond the bound A, are greater than the chosen bound K. In other words, the function f(x) grows without bound. Whatever bound we choose on the y-axis, the function eventually exceeds it.

You can picture the simple function f(x) = x. If we choose a bound K on the y-axis, then for all x > K we get function values that are greater than K.