Question

what is difference between f(1) and lim f(x) as x approaches 1 ?

Answer 1

f(1) is the value at the exact location of 1
lim x->1 f(x) is what the values are going to as you approach infinitely closer to 1 from either side of it.

Answer 2

consider f(x) = 1/(x-1)
what is f(1)?
it doesn't exist

what is lim x->1 f(x)?
it is negative infinity from the left, and pos inf from the right.