Proving delta epsilon limits at infinity for unknown function

Question

I have a question and it's limx>inf (f(x)= 2 and I have to prove that N exists and if x>N 1<f(x)<3

I started trying to set up my proof with |f(x)-2|<E but I've never done a problem like this with an unknown function and I have no idea what to do next. If anyone can help solve this or let me know where to start, it would be greatly appreciated!

Answer 1

Here is one way to word this problem.

We are given that lim (x->∞) (f(x)) = 2. This means that for any positive number ε, there exists a number N_ε, such that when

x > N_ε,

then

|f(x) - 2| < ε.

Since the above statement is true, we know, that there is a positive number N₁, such that

when

x > N₁

then

|f(x) - 2| < 1.

With other words,

-1 < f(x) - 2 < 1,

1 < f(x) < 3.