f(x) = x / (x² - x) lim f(x) = x→1

Explanation:

Step1: Simplify the function

First, factor the denominator of $f(x)=\frac{x}{x^{2}-x}$. We have $x^{2}-x=x(x - 1)$, so $f(x)=\frac{x}{x(x - 1)}=\frac{1}{x - 1},x\neq0,x\neq1$.

Step2: Calculate the limit

We want to find $\lim_{x\rightarrow1}\frac{1}{x - 1}$. As $x\rightarrow1$, the denominator $x - 1\rightarrow0$. When $x$ approaches $1$ from the left - hand side ($x\lt1$), $x - 1\lt0$ and $\frac{1}{x - 1}\rightarrow-\infty$. When $x$ approaches $1$ from the right - hand side ($x\gt1$), $x - 1\gt0$ and $\frac{1}{x - 1}\rightarrow+\infty$. Since the left - hand limit and the right - hand limit are not equal, the limit $\lim_{x\rightarrow1}f(x)$ does not exist.

Answer:

The limit does not exist.