Infinite function must be unbounded, but why unbounded function is not necessarily infinite?

Infinite function must be unbounded, but why unbounded function is not necessarily infinite?

Unbounded function may have subsequence and subsequence has limit, then it is not infinite
For example, f (x) = xcosx is unbounded in (- ∞, + ∞), but not infinite when x → + ∞
There exists a sequence xn = 2n π, f (xn) = 2n π → + ∞ (n →∞), so {f (xn)} is unbounded, so the function f (x) is unbounded in (- ∞, + ∞)
There exists the sequence yn = 2n π + π / 2, f (yn) = 0, so the function f (x) is not infinite when x → + ∞

What is the difference between unbounded function and infinite function when x →∞?

When x tends to infinity, the definition of infinity is: for any M > 0, there exists x > 0, such that when | x | > x, there is | f (x) | > m; for unbounded, it can be defined according to the definition of bounded and the duality rule: for any m > 0, there exists x, such that | f (x) | > M

How to distinguish infinity, boundedness, unbounded and limit of function?
Does boundedness mean limit?
And infinity, isn't it: unbounded, so infinity?
Look at this topic: is it bounded or infinite? Why···

Infinity: more and more big, endless big down, infinitely big down. However, can not be between positive and negative infinity fluctuations
Bounded: there is a range limiting the range of a function
Unbounded: there is no scope to limit, one moment to positive infinity, one moment to negative infinity
Limit: more and more tend to a fixed value, function value and fixed value of the absolute value of the difference tends to infinitesimal
Exception: if monotonically tends to positive infinity, we also say that the limit is positive infinity,
If monotonically tends to negative infinity, we also say that the limit is negative infinity
But if one moment is positive and the other is negative, the absolute value tends to infinity,
That is, between positive and negative infinity fluctuation, we say "limit does not exist."
When x tends to 0, 1 / X & sup2; tends to infinity; sin (1 / x) is bounded, between ± 1, but not infinity