Limit of functions and sequences

Limit is used to describe that in function F(x), if x becomes bigger, F(x) is approaching a constant.
However, in most cases y will only be very close to the constant and never really equals it. As the x is bigger, the y is closer to the constant, so we assume that when x reaches infinity, y will reach the constant.
That is quite reasonable because both infinity and the constant can't really be reached in real world.
Some people may not agree this because if the limit is constant, it is to say that 0.9999(recurring decimal)=1 as they are as close as possible.

In fact, the limit also has a strong relationship with sequences.
Assume there is an infinity sequence {an}(n∈R). When the item approaches a constant A, then A is the limit of this sequence, lim an=A.