This is a topic about the relationship between function limit and sequence limit.

This is a theorem. If lim(x→x0)f(x) exists, {xn} is an arbitrary convergent sequence of x0 in the definition domain of function f(x), and it is satisfied that xn is not equal to x0(n belongs to Z+), then the corresponding sequence of function values {f (xn)} must converge.

And lim(n→∞)f(xn)= lim(x→x0)f(x).

Understanding: In the sequence, when n tends to ∝, xn changes, (note that xn is not equal to x0), and xn changes lead to f(xn) changes.

This sentence can also be interpreted as, in the function, the change of x tends to x0, which leads to the change of f(x), so we can draw a conclusion.

lim(n→∞)f(xn)= lim(x→x0)f(x)