Calculus Examples

$lim_{x \to 3^{-}} \frac{9 - x^{2}}{4 x | x - 3 |}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 3^{-}} \frac{9 - x^{2}}{x | x - 3 |}$

Step 2

Make a table to show the behavior of the function $\frac{9 - x^{2}}{x | x - 3 |}$ as $x$ approaches $3$ from the left.

$\begin{array}{c} x & \frac{9 - x^{2}}{x | x - 3 |} \\ 2.9 & 2.03448275 \\ 2.99 & 2.00334448 \\ 2.999 & 2.00033344 \\ 2.9999 & 2.00003333 \end{array}$

Step 3

As the $x$ values approach $3$ , the function values approach $2$ . Thus, the limit of $\frac{9 - x^{2}}{x | x - 3 |}$ as $x$ approaches $3$ from the left is $2$ .

$\frac{1}{4} \cdot 2$

Step 4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} \cdot 2$

Step 4.2

Cancel the common factor.

$\frac{1}{2 \cdot 2} \cdot 2$

Step 4.3

Rewrite the expression.

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$