Asymptotes are the places where x makes the denominator equal to 0. That means that if we are given 5/(x-4), when x=4, then the denominator becomes 5/0 We cannot divide by 0 without destroying the universe or ripping a hole in the fabric of reality. All in all, we just stay away from dividing by 0. Since our calculators, and the mathmaticians before them, can't handle the whole concept of 5/0, we just skip over it. That is what an asyptoote it. The graph may get very, very, very, very, very close, but x willl never equal 4.
In our case, we have 1/(x-1)^2, or 1/((x-1)(x-1)). We need to find the value that will make x equal to 0. We do that by saying 0=x-1 and solving for x, like this: x=1. So, we now that when x=1, the graph freaks out. that is our asymptote. It makes no difference that there are two values that the same; it just changes the shape of the grapg. Let's graph it and look at what we've got:
graph{y=1/(x-1)^2}
