Answer
The quotient property of limits cannot be used when the limit of the denominator is zero.
Work Step by Step
Consider the limit $\underset{x\to 3}{\mathop{\lim }}\,\frac{{{x}^{2}}-x-6}{x-3}$.
In this limit, the limit of the denominator is zero so the quotient property of limits cannot be used.
This is because, on applying the quotient property,
$\begin{align}
& \underset{x\to 3}{\mathop{\lim }}\,\frac{{{x}^{2}}-x-6}{x-3}=\frac{\underset{x\to 3}{\mathop{\lim }}\,\left( {{x}^{2}}-x-6 \right)}{\underset{x\to 3}{\mathop{\lim }}\,\left( x-3 \right)} \\
& =\frac{{{3}^{2}}-3-6}{3-3} \\
& =\frac{9-9}{3-3} \\
& =\frac{0}{0}
\end{align}$
And any number divided by zero is undefined.
So, the quotient property of limits cannot be applied on those limit of quotients in which the limit of the denominator is zero.
