If there is no limit, is it infinite? I don't think so. Please explain why. Thank you

If there is no limit, is it infinite? I don't think so. Please explain why. Thank you

For example, SiNx, X - > infinity

High number problem: what are the definitions of infinite number and unbounded variable? Why is it said that infinite number must be unbounded variable, but unbounded variable is not necessarily infinite
High number problem: what are the definitions of infinite and unbounded variables?
Why is it said that an infinite number must be an unbounded variable, but an unbounded variable is not necessarily an infinite number?

Definition 1: if for any given positive number m, there exists δ > 0 (or positive number x), so that when 0 M. obviously, the even number item in the above sequence can not meet this requirement

Higher numbers, on bounded problems
A sequence is bounded. In the definition, it is said that there is a positive number m such that | xn | a (a is a constant greater than zero), so the sequence is bounded. Isn't this contrary to the definition? In the definition, it is | xn | a, which is greater than a positive number. Why? Is there a lower bound also bounded?
This problem is: let a > 0, X1 > 0, xn + 1 = 1 / 2 (xn + A / xn), (n = 1,2,3...). 1. Prove that the sequence {xn} monotonically decreases and has lower bound. 2.lim Xn
n－＞∞
Although the first question is to prove that there is a lower bound. But the second question is to find the limit. The answer given in the book is: from the first question, we can see that the limit exists. What he means is that if there is a lower bound, there will be a bound. Otherwise, how can we know the existence of limit from the first question?

The proof of monotone bounded theorem with limit means that monotone increasing has upper bound or monotone decreasing has lower bound
As we said on the first floor, when we say that a sequence is bounded, we mean that the sequence has both upper and lower bounds, | xn|