Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $8 π$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

$8 \cdot \sin (8 π \frac{1}{8})$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $8$ .

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Step 7.1.1

Factor $8$ out of $8 π$ .

$8 \cdot \sin (8 (π) \frac{1}{8})$

Step 7.1.2

Cancel the common factor.

$8 \cdot \sin (8 π \frac{1}{8})$

Step 7.1.3

Rewrite the expression.

$8 \cdot \sin (π)$

Step 7.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$8 \cdot \sin (0)$

Step 7.3

The exact value of $\sin (0)$ is $0$ .

$8 \cdot 0$

Step 7.4

Multiply $8$ by $0$ .

$0$