Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 41

Determine the limit $\lim\limits_{x \rightarrow 0.5^-} \displaystyle \frac{2x-1}{|2x^3-x^2|}$, if it exists. If the limit does not exist, explain why.


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\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow 0.5^-} \displaystyle \frac{2x-1}{|2x^3-x^2|} & = \lim\limits_{x \rightarrow 0.5^-} \displaystyle \left[ \frac{2x-1}{-(2x^3-x^2)}\right]
= \lim\limits_{x \rightarrow 0.5^-} \displaystyle \frac{\cancel{2x-1}}{-\cancel{(2x-1)}(x^2)} && \text{(Get the factor and simplify)}\\
\lim\limits_{x \rightarrow 0.5^-} \displaystyle \frac{1}{-x^2} & = -\frac{1}{(0.5)^2} && \text{(Substitute value of } x)
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \rightarrow 0.5^-} \displaystyle \frac{2x-1}{|2x^3-x^2|} = -4}
$