Calculus Examples

$lim_{x \to 0.5} \frac{2 x - 1}{| 2 x^{3} - x^{2} |}$

Step 1

Change the two-sided limit into a left sided limit.

$lim_{x \to {0.5}^{-}} \frac{2 x - 1}{| 2 x^{3} - x^{2} |}$

Step 2

Make a table to show the behavior of the function $\frac{2 x - 1}{| 2 x^{3} - x^{2} |}$ as $x$ approaches $0.5$ from the left.

$\begin{array}{c} x & \frac{2 x - 1}{| 2 x^{3} - x^{2} |} \\ 0.4 & - 6.25 \\ 0.49 & - 4.16493127 \\ 0.499 & - 4.01604812 \\ 0.4999 & - 4.00160048 \\ 0.49999 & - 4.00016 \\ 0.499999 & - 4.000016 \end{array}$

Step 3

As the $x$ values approach $0.5$ , the function values approach $- 4$ . Thus, the limit of $\frac{2 x - 1}{| 2 x^{3} - x^{2} |}$ as $x$ approaches $0.5$ from the left is $- 4$ .

$- 4$