Calculus Examples

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

Step 1

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

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Step 1.1

Factor $3$ out of $6 \sin (x)$ .

$lim_{x \to 0^{+}} \frac{3 (2 \sin (x))}{9 | x |}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $9 | x |$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

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 right is $1$ .

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

Step 5

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$