Calculus Examples

$lim_{x \to \frac{1}{6}} {(6 \sin (π x) - 1)}^{3}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $3$ from ${(6 \sin (π x) - 1)}^{3}$ outside the limit using the Limits Power Rule.

${(lim_{x \to \frac{1}{6}} 6 \sin (π x) - 1)}^{3}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{6}$ .

${(lim_{x \to \frac{1}{6}} 6 \sin (π x) - lim_{x \to \frac{1}{6}} 1)}^{3}$

Step 1.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

${(6 lim_{x \to \frac{1}{6}} \sin (π x) - lim_{x \to \frac{1}{6}} 1)}^{3}$

Step 1.4

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

${(6 \sin (lim_{x \to \frac{1}{6}} π x) - lim_{x \to \frac{1}{6}} 1)}^{3}$

Step 1.5

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

${(6 \sin (π lim_{x \to \frac{1}{6}} x) - lim_{x \to \frac{1}{6}} 1)}^{3}$

Step 1.6

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{6}$ .

${(6 \sin (π lim_{x \to \frac{1}{6}} x) - 1 \cdot 1)}^{3}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{6}$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine $π$ and $\frac{1}{6}$ .

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

Step 3.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

${(6 (\frac{1}{2}) - 1 \cdot 1)}^{3}$

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Factor $2$ out of $6$ .

${(2 (3) \frac{1}{2} - 1 \cdot 1)}^{3}$

Step 3.1.3.2

Cancel the common factor.

${(2 \cdot 3 \frac{1}{2} - 1 \cdot 1)}^{3}$

Step 3.1.3.3

Rewrite the expression.

${(3 - 1 \cdot 1)}^{3}$

Step 3.1.4

Multiply $- 1$ by $1$ .

${(3 - 1)}^{3}$

Step 3.2

Subtract $1$ from $3$ .

$2^{3}$

Step 3.3

Raise $2$ to the power of $3$ .

$8$