Calculus Examples

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

Step 1

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

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

Step 2

Move the exponent $2$ from $n^{2}$ outside the limit using the Limits Power Rule.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

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 $n$ .

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Raise $8$ to the power of $2$ .

$64 \cdot \sin (\frac{1}{2^{8}})$

Step 8.2

Raise $2$ to the power of $8$ .

$64 \cdot \sin (\frac{1}{256})$

Step 8.3

Evaluate $\sin (\frac{1}{256})$ .

$64 \cdot 0.00006817$

Step 8.4

Multiply $64$ by $0.00006817$ .

$0.00436332$