Calculus Examples

$lim_{x \to 8} \cos (\frac{π}{x})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\cos (lim_{x \to 8} \frac{π}{x})$

Step 1.2

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

$\cos (π lim_{x \to 8} \frac{1}{x})$

Step 1.3

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

$\cos (π \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 1.4

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

$\cos (π \frac{1}{lim_{x \to 8} x})$

Step 2

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

$\cos (π \frac{1}{8})$

Step 3

Simplify the answer.

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Step 3.1

Combine $π$ and $\frac{1}{8}$ .

$\cos (\frac{π}{8})$

Step 3.2

The exact value of $\cos (\frac{π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 3.2.1

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

$\cos (\frac{\frac{π}{4}}{2})$

Step 3.2.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\pm \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}}$

Step 3.2.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$\sqrt{\frac{1 + \cos (\frac{π}{4})}{2}}$

Step 3.2.4

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

$\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

Step 3.2.5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

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Step 3.2.5.1

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

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}$

Step 3.2.5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}$

Step 3.2.5.3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}}$

Step 3.2.5.4

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

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Step 3.2.5.4.1

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

$\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}}$

Step 3.2.5.4.2

Multiply $2$ by $2$ .

$\sqrt{\frac{2 + \sqrt{2}}{4}}$

Step 3.2.5.5

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

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$

Step 3.2.5.6

Simplify the denominator.

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Step 3.2.5.6.1

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

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}}$

Step 3.2.5.6.2

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

$\frac{\sqrt{2 + \sqrt{2}}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2 + \sqrt{2}}}{2}$

Decimal Form:

$0.92387953 \dots$