Calculus Examples

$lim_{x \to 0^{-}} \frac{9 \sin (x)}{6 | x |}$

Step 1

Cancel the common factor of $9$ and $6$ .

Tap for more steps...

Step 1.1

Factor $3$ out of $9 \sin (x)$ .

$lim_{x \to 0^{-}} \frac{3 (3 \sin (x))}{6 | x |}$

Step 1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1

Factor $3$ out of $6 | x |$ .

$lim_{x \to 0^{-}} \frac{3 (3 \sin (x))}{3 (2 | x |)}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to 0^{-}} \frac{3 (3 \sin (x))}{3 (2 | x |)}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to 0^{-}} \frac{3 \sin (x)}{2 | x |}$

Step 2

Move the term $\frac{3}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{2} lim_{x \to 0^{-}} \frac{\sin (x)}{| x |}$

Step 3

Make a table to show the behavior of the function $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \frac{\sin (x)}{| x |} \\ - 0.1 & - 0.99833416 \\ - 0.01 & - 0.99998333 \\ - 0.001 & - 0.99999983 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $- 1$ . Thus, the limit of $\frac{\sin (x)}{| x |}$ as $x$ approaches $0$ from the left is $- 1$ .

$\frac{3}{2} \cdot - 1$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Multiply $\frac{3}{2} \cdot - 1$ .

Tap for more steps...

Step 5.1.1

Combine $\frac{3}{2}$ and $- 1$ .

$\frac{3 \cdot - 1}{2}$

Step 5.1.2

Multiply $3$ by $- 1$ .

$\frac{- 3}{2}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$