Calculus Examples

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

Step 1

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

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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.

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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 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 $1$ .

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

Step 5

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

$\frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$