Calculus Examples

$lim_{x \to 0} x^{2} \sin (x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} x^{2} \cdot lim_{x \to 0} \sin (x)$

Step 2

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

${(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} \sin (x)$

Step 3

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

${(lim_{x \to 0} x)}^{2} \cdot \sin (lim_{x \to 0} x)$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0^{2} \cdot \sin (lim_{x \to 0} x)$

Step 4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0^{2} \cdot \sin (0)$

Step 5

Simplify the answer.

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Step 5.1

Raising $0$ to any positive power yields $0$ .

$0 \cdot \sin (0)$

Step 5.2

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

$0 \cdot 0$

Step 5.3

Multiply $0$ by $0$ .

$0$