# How do you find the asymptotes of  (1) / (x-1) ^ 2?

The asymptotes are $x = 1$.
Asymptotes are the places where $x$ makes the denominator equal to $0$. That means that if we are given $\frac{5}{x - 4}$, when $x = 4$, then the denominator becomes $\frac{5}{0}$ 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 $\frac{5}{0}$, 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 $\frac{1}{x - 1} ^ 2$, or $\frac{1}{\left(x - 1\right) \left(x - 1\right)}$. 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: