Calculus Examples

$lim_{x \to \frac{π}{6}} \frac{x}{\sin (3 x)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{6}$ .

$\frac{lim_{x \to \frac{π}{6}} x}{lim_{x \to \frac{π}{6}} \sin (3 x)}$

Step 2

Move the limit inside the trig function because sine is continuous.

$\frac{lim_{x \to \frac{π}{6}} x}{\sin (lim_{x \to \frac{π}{6}} 3 x)}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to \frac{π}{6}} x}{\sin (3 lim_{x \to \frac{π}{6}} x)}$

Step 4

Evaluate the limits by plugging in $\frac{π}{6}$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{\frac{π}{6}}{\sin (3 lim_{x \to \frac{π}{6}} x)}$

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{\frac{π}{6}}{\sin (3 \frac{π}{6})}$

Step 5

Simplify the answer.

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Step 5.1

Separate fractions.

$\frac{\frac{π}{6}}{1} \cdot \frac{1}{\sin (3 \frac{π}{6})}$

Step 5.2

Convert from $\frac{1}{\sin (3 \frac{π}{6})}$ to $\csc (3 \frac{π}{6})$ .

$\frac{\frac{π}{6}}{1} \csc (3 \frac{π}{6})$

Step 5.3

Divide $\frac{π}{6}$ by $1$ .

$\frac{π}{6} \csc (3 \frac{π}{6})$

Step 5.4

Cancel the common factor of $3$ .

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Step 5.4.1

Factor $3$ out of $6$ .

$\frac{π}{6} \csc (3 \frac{π}{3 (2)})$

Step 5.4.2

Cancel the common factor.

$\frac{π}{6} \csc (3 \frac{π}{3 \cdot 2})$

Step 5.4.3

Rewrite the expression.

$\frac{π}{6} \csc (\frac{π}{2})$

Step 5.5

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$\frac{π}{6} \cdot 1$

Step 5.6

Multiply $\frac{π}{6}$ by $1$ .

$\frac{π}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{6}$

Decimal Form:

$0.52359877 \dots$