Calculus Examples

$lim_{x \to 8} (\sqrt{x}) \sin (\frac{1}{x})$

Step 1

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

$lim_{x \to 8} \sqrt{x} \cdot lim_{x \to 8} \sin (\frac{1}{x})$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x} \cdot lim_{x \to 8} \sin (\frac{1}{x})$

Step 3

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

$\sqrt{lim_{x \to 8} x} \cdot \sin (lim_{x \to 8} \frac{1}{x})$

Step 4

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

$\sqrt{lim_{x \to 8} x} \cdot \sin (\frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 5

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

$\sqrt{lim_{x \to 8} x} \cdot \sin (\frac{1}{lim_{x \to 8} x})$

Step 6

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

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Step 6.1

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

$\sqrt{8} \cdot \sin (\frac{1}{lim_{x \to 8} x})$

Step 6.2

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

$\sqrt{8} \cdot \sin (\frac{1}{8})$

Step 7

Simplify the answer.

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Step 7.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.1

Factor $4$ out of $8$ .

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

Step 7.1.2

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

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

Step 7.2

Pull terms out from under the radical.

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

Step 7.3

Evaluate $\sin (\frac{1}{8})$ .

$2 \sqrt{2} \cdot 0.12467473$

Step 7.4

Multiply $2 \sqrt{2} \cdot 0.12467473$ .

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Step 7.4.1

Multiply $0.12467473$ by $2$ .

$0.24934946 \sqrt{2}$

Step 7.4.2

Multiply $0.24934946$ by $\sqrt{2}$ .

$0.35263339$