Calculus Examples

$lim_{x \to 8} x \sin (\frac{9 π}{x})$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot lim_{x \to 8} \sin (\frac{9 π}{x})$

Step 2

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

$lim_{x \to 8} x \cdot \sin (lim_{x \to 8} \frac{9 π}{x})$

Step 3

Move the term $9 π$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 8} x \cdot \sin (9 π lim_{x \to 8} \frac{1}{x})$

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot \sin (9 π \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$lim_{x \to 8} x \cdot \sin (9 π \frac{1}{lim_{x \to 8} x})$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$8 \cdot \sin (9 π \frac{1}{lim_{x \to 8} x})$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$8 \cdot \sin (9 π \frac{1}{8})$

Step 7

Simplify the answer.

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Step 7.1

Multiply $9 π \frac{1}{8}$ .

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Step 7.1.1

Combine $\frac{1}{8}$ and $9$ .

$8 \cdot \sin (\frac{9}{8} π)$

Step 7.1.2

Combine $\frac{9}{8}$ and $π$ .

$8 \cdot \sin (\frac{9 π}{8})$

Step 7.2

The exact value of $\sin (\frac{9 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 7.2.1

Rewrite $\frac{9 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$8 \cdot \sin (\frac{\frac{9 π}{4}}{2})$

Step 7.2.2

Apply the sine half-angle identity.

$8 \cdot (\pm \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}})$

Step 7.2.3

Change the $\pm$ to $-$ because sine is negative in the third quadrant.

$8 \cdot (- \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}})$

Step 7.2.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{9 π}{4})}{2}}$ .

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Step 7.2.4.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$8 \cdot (- \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}})$

Step 7.2.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$8 \cdot (- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}})$

Step 7.2.4.3

Write $1$ as a fraction with a common denominator.

$8 \cdot (- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}})$

Step 7.2.4.4

Combine the numerators over the common denominator.

$8 \cdot (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}})$

Step 7.2.4.5

Multiply the numerator by the reciprocal of the denominator.

$8 \cdot (- \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}})$

Step 7.2.4.6

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

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Step 7.2.4.6.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$8 \cdot (- \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}})$

Step 7.2.4.6.2

Multiply $2$ by $2$ .

$8 \cdot (- \sqrt{\frac{2 - \sqrt{2}}{4}})$

Step 7.2.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$8 \cdot (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}})$

Step 7.2.4.8

Simplify the denominator.

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Step 7.2.4.8.1

Rewrite $4$ as $2^{2}$ .

$8 \cdot (- \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}})$

Step 7.2.4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$8 \cdot (- \frac{\sqrt{2 - \sqrt{2}}}{2})$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Move the leading negative in $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ into the numerator.

$8 \cdot \frac{- \sqrt{2 - \sqrt{2}}}{2}$

Step 7.3.2

Factor $2$ out of $8$ .

$2 (4) \cdot \frac{- \sqrt{2 - \sqrt{2}}}{2}$

Step 7.3.3

Cancel the common factor.

$2 \cdot 4 \cdot \frac{- \sqrt{2 - \sqrt{2}}}{2}$

Step 7.3.4

Rewrite the expression.

$4 \cdot (- \sqrt{2 - \sqrt{2}})$

Step 7.4

Multiply $- 1$ by $4$ .

$- 4 \sqrt{2 - \sqrt{2}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- 4 \sqrt{2 - \sqrt{2}}$

Decimal Form:

$- 3.06146745 \dots$