Answer
$-1$
Work Step by Step
We want to find $\lim\limits_{x \to 1}\dfrac{\frac{1}{x}-1}{x-1}$, but we can't use the quotient rule for limits because the limit of the denominator as $x$ approaches 1 is zero, and if we try to substitute $x=1$ directly, we get zero in both the numerator and the denominator.
Note that $\dfrac{\frac{1}{x}-1}{x-1}=\left(\dfrac{\frac{1}{x}-1}{x-1}\right)\left(\dfrac{x}{x}\right)=\dfrac{1-x}{x(x-1)}=\dfrac{-(x-1)}{x(x-1)}=\dfrac{-1}{x}$
for all $x \neq 1.$
Thus $\lim\limits_{x \to 1}\dfrac{\frac{1}{x}-1}{x-1}=\lim\limits_{x \to 1}\dfrac{-1}{x}=\dfrac{-1}{1}=-1.$
