Calculus Examples

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

Step 1

Move the limit inside the absolute value signs.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

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

$| 64 \cdot 0.12467473 |$

Step 8.3

Multiply $64$ by $0.12467473$ .

$| 7.97918293 |$

Step 8.4

The absolute value is the distance between a number and zero. The distance between $0$ and $7.97918293$ is $7.97918293$ .

$7.97918293$