Calculus Examples

$\frac{lim_{x \to 0} x \sin (x)}{3 - 3 \cos (x)}$

Step 1

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

$\frac{lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x)}{3 - 3 \cos (x)}$

Step 2

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

$\frac{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}{3 - 3 \cos (x)}$

Step 3

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

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Step 3.1

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

$\frac{0 \cdot \sin (lim_{x \to 0} x)}{3 - 3 \cos (x)}$

Step 3.2

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

$\frac{0 \cdot \sin (0)}{3 - 3 \cos (x)}$

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $0$ and $3 - 3 \cos (x)$ .

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Step 4.1.1

Factor $3$ out of $0 \cdot \sin (0)$ .

$\frac{3 (0 \cdot \sin (0))}{3 - 3 \cos (x)}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $3$ out of $3$ .

$\frac{3 (0 \cdot \sin (0))}{3 (1) - 3 \cos (x)}$

Step 4.1.2.2

Factor $3$ out of $- 3 \cos (x)$ .

$\frac{3 (0 \cdot \sin (0))}{3 (1) + 3 (- \cos (x))}$

Step 4.1.2.3

Factor $3$ out of $3 (1) + 3 (- \cos (x))$ .

$\frac{3 (0 \cdot \sin (0))}{3 (1 - \cos (x))}$

Step 4.1.2.4

Cancel the common factor.

$\frac{3 (0 \cdot \sin (0))}{3 (1 - \cos (x))}$

Step 4.1.2.5

Rewrite the expression.

$\frac{0 \cdot \sin (0)}{1 - \cos (x)}$

Step 4.2

Multiply $0$ by $\sin (0)$ .

$\frac{0}{1 - \cos (x)}$

Step 4.3

Divide $0$ by $1 - \cos (x)$ .

$0$