Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Multiply $\frac{π}{2} \cdot \frac{1}{8}$ .

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Step 7.1.1

Multiply $\frac{π}{2}$ by $\frac{1}{8}$ .

$8 \cdot \sin (\frac{π}{2 \cdot 8})$

Step 7.1.2

Multiply $2$ by $8$ .

$8 \cdot \sin (\frac{π}{16})$

Step 7.2

Evaluate $\sin (\frac{π}{16})$ .

$8 \cdot 0.19509032$

Step 7.3

Multiply $8$ by $0.19509032$ .

$1.56072257$