An eigenlimit at an eigenpoint

The limit of a function is one of the most important concepts in mathematical analysis. It describes the behaviour of a function around a certain point, which allows us to define, for example, the continuity of a function. The limit of a function helps us understand the behaviour of a function even at points where it is not defined at all.

Non-eigenlimit at a non-eigenpoint

A combination of the previous limits - we are looking for the limit of a function for x approaching plus or minus infinity, and the limit itself will also come out either plus or minus infinity.

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

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

and at the point −∞ if

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

Similarly for x approaching −∞. The definition combines the previous principles. It tells us that if we are looking for a limit at infinity, which in turn is supposed to be infinity, then for every limit K on the y axis we find a limit A on the x axis such that all function values f(x) for x > A, i.e., beyond the limit A, are greater than the chosen limit K. In other words, the function f(x) grows beyond all limits. Whatever limit we choose on the y axis, the function will always outgrow it over time.

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