Calculus Examples

lim x \to 2^{-} \frac{∣ ∣ 2 x - x^{2} ∣ ∣}{x - 2}

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

Step 1

Make a table to show the behavior of the function

\frac{∣ ∣ 2 x - x^{2} ∣ ∣}{x - 2}

$\frac{| 2 x - x^{2} |}{x - 2}$ as

x

$x$ approaches

2

$2$ from the left.

\begin{matrix} x & \frac{∣ ∣ 2 x - x^{2} ∣ ∣}{x - 2} 1.9 & - 1.9 1.99 & - 1.99 1.999 & - 1.999 1.9999 & - 1.9999 1.99999 & - 1.99999 \end{matrix}

$\begin{array}{c} x & \frac{| 2 x - x^{2} |}{x - 2} \\ 1.9 & - 1.9 \\ 1.99 & - 1.99 \\ 1.999 & - 1.999 \\ 1.9999 & - 1.9999 \\ 1.99999 & - 1.99999 \end{array}$

Step 2

As the

x

$x$ values approach

2

$2$ , the function values approach

- 2

$- 2$ . Thus, the limit of

\frac{∣ ∣ 2 x - x^{2} ∣ ∣}{x - 2}

$\frac{| 2 x - x^{2} |}{x - 2}$ as

x

$x$ approaches

2

$2$ from the left is

- 2

$- 2$ .

- 2

$- 2$