Calculus Examples

$lim_{n \to 8} \sin (\frac{2 π}{n})$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\sin (lim_{n \to 8} \frac{2 π}{n})$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

$\sin (2 π \frac{1}{8})$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.1

Factor $2$ out of $2 π$ .

$\sin (2 (π) \frac{1}{8})$

Step 3.1.2

Factor $2$ out of $8$ .

$\sin (2 (π) \frac{1}{2 (4)})$

Step 3.1.3

Cancel the common factor.

$\sin (2 π \frac{1}{2 \cdot 4})$

Step 3.1.4

Rewrite the expression.

$\sin (π \frac{1}{4})$

Step 3.2

Combine $π$ and $\frac{1}{4}$ .

$\sin (\frac{π}{4})$

Step 3.3

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$