You want the limit as x->2, so you are concerned with x of the form 2+d for small values of d. The question is how small.
Start by substituting for x. x2-3 becomes d2+4d+1. Since we are interested in the difference of this from one, we can subtract one and see how small d must be to make what's left small enough.
Subtracting one leaves us with d2+4d. And that must be made smaller than e in magnitude. We can factor this as d(d+4). If d is greater than zero, then clearly this difference is also greater than zero and we need an upper limit for it. Further if d<1 then d(d+4)<5d. And if d<e/5 then 5d<e
So if d is less than both one and e/5 we will have (x2-3)-1 less than e. You can work out a similar set of inequalities for d (and the difference) less than zero, and combine them to get inequalities in the absolute values.