Do you really need to apply a proof here?
The definition of "limit" says that, given any , there exist such that if then .
Here, you are given that so we want to have
Go ahead and add the fractions on the left:
So we want .
That is the same as
Let's say that |5- x|< 1. Then -1< x- 5< 1 so 4< x< 6. then 2< x- 2< 4 so 2< |x-2|< 4. Certainly, then, so if we can make we are done. Since we are already assuming , and , it is sufficient to make . That is, we can take .
Notice that this is not saying that has to be equal to 1. Obviously any smaller number would work and there might be some larger value that would work. But is sufficient.
Had I said "Let's say that |5- x|<" some other number, I would have gotten an answer other than 1, but equally valid.