Calculus Examples

$lim_{n \to 8} (\frac{n}{2 π}) \sin (\frac{2 π}{n}) = 1$

Step 1

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

$lim_{n \to 8} \frac{n}{2 π} \cdot lim_{n \to 8} \sin (\frac{2 π}{n}) = 1$

Step 2

Move the term $\frac{1}{2 π}$ outside of the limit because it is constant with respect to $n$ .

$\frac{1}{2 π} lim_{n \to 8} n \cdot lim_{n \to 8} \sin (\frac{2 π}{n}) = 1$

Step 3

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

$\frac{1}{2 π} lim_{n \to 8} n \cdot \sin (lim_{n \to 8} \frac{2 π}{n}) = 1$

Step 4

Move the term $2 π$ outside of the limit because it is constant with respect to $n$ .

$\frac{1}{2 π} lim_{n \to 8} n \cdot \sin (2 π lim_{n \to 8} \frac{1}{n}) = 1$

Step 5

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

$\frac{1}{2 π} lim_{n \to 8} n \cdot \sin (2 π \frac{lim_{n \to 8} 1}{lim_{n \to 8} n}) = 1$

Step 6

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

$\frac{1}{2 π} lim_{n \to 8} n \cdot \sin (2 π \frac{1}{lim_{n \to 8} n}) = 1$

Step 7

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

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Step 7.1

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

$\frac{1}{2 π} \cdot 8 \cdot \sin (2 π \frac{1}{lim_{n \to 8} n}) = 1$

Step 7.2

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

$\frac{1}{2 π} \cdot 8 \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $2 π$ .

$\frac{1}{2 (π)} \cdot 8 \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8.1.2

Factor $2$ out of $8$ .

$\frac{1}{2 (π)} \cdot (2 (4)) \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8.1.3

Cancel the common factor.

$\frac{1}{2 π} \cdot (2 \cdot 4) \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8.1.4

Rewrite the expression.

$\frac{1}{π} \cdot 4 \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8.2

Combine $\frac{1}{π}$ and $4$ .

$\frac{4}{π} \cdot \sin (2 π \frac{1}{8}) = 1$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $2 π$ .

$\frac{4}{π} \cdot \sin (2 (π) \frac{1}{8}) = 1$

Step 8.3.2

Factor $2$ out of $8$ .

$\frac{4}{π} \cdot \sin (2 (π) \frac{1}{2 (4)}) = 1$

Step 8.3.3

Cancel the common factor.

$\frac{4}{π} \cdot \sin (2 π \frac{1}{2 \cdot 4}) = 1$

Step 8.3.4

Rewrite the expression.

$\frac{4}{π} \cdot \sin (π \frac{1}{4}) = 1$

Step 8.4

Combine $π$ and $\frac{1}{4}$ .

$\frac{4}{π} \cdot \sin (\frac{π}{4}) = 1$

Step 8.5

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

$\frac{4}{π} \cdot \frac{\sqrt{2}}{2} = 1$

Step 8.6

Cancel the common factor of $2$ .

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Step 8.6.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{π} \cdot \frac{\sqrt{2}}{2} = 1$

Step 8.6.2

Cancel the common factor.

$\frac{2 \cdot 2}{π} \cdot \frac{\sqrt{2}}{2} = 1$

Step 8.6.3

Rewrite the expression.

$\frac{2}{π} \cdot \sqrt{2} = 1$

Step 8.7

Combine $\frac{2}{π}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{π} = 1$