I'm not sure I understand your question. is a rational function so that it is "continuous" where it is defined. In particular, setting x= 1, . Since that exists, the function is continuous and its limit, as x goes to 1, is the value of the function, 1. And since the limit exists, the two one sided limits must be the same, 1.
If you don't like that, you could copy the proof of the limit, restricting x to be larger than 2: Given , we need to find , such that if (we don't need the absolute value because we are assuming x> 2) then . so we have . If x is larger than 1 but close to it, we can assume 1/2< x< 3/2 so that 1/2- 2= -3/2< x- 2< 3/2- 2= -1/2 and |x- 2|> 1/2. With that, if then .