Answer:

It is not unreasonable to say that as x approaches 1 the functions f[x] and x+1 have the same behavior, so we expect that f[x] and (x+1) to be the same. It is clear that if x is getting closer and closer to 1, then the value of x+1 must be getting closer and closer to 1 + 1 = 2.
Here are Mathematica 's computations of the two limits.
First, f[x]:

Limit[f[x],x->1]

And now (x+1):

Limit[x+1,x->1]