Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Factor $2$ out of $8$ .

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

Step 7.1.2

Cancel the common factor.

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

Step 7.1.3

Rewrite the expression.

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

Step 7.2

Evaluate $\sin (\frac{1}{4})$ .

$8 \cdot 0.24740395$

Step 7.3

Multiply $8$ by $0.24740395$ .

$1.97923167$