Answer
The limit does not exist because $f(x)$ grows too large to have a limit as $x\to1$
Work Step by Step
$$\lim_{x\to1}\frac{1}{x-1}$$
As $x\to1$, $(x-1)$ approaches $0$, so $\frac{1}{x-1}$ would go infinitely large and does not revolve around any fixed real number. And no, $\infty$ is not accepted as a limit.
In other words, the function is not bounded, so its limit does not exist as $x\to 1$.
A graph has been enclosed below, showing the graph as $x\to1$.
