Calculus Examples

$\frac{lim_{x \to 3} \sin (x - 3)}{3 x - 9}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 3} x - 3)}{3 x - 9}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{\sin (lim_{x \to 3} x - lim_{x \to 3} 3)}{3 x - 9}$

Step 1.3

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3)}{3 x - 9}$

Step 2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\sin (3 - 1 \cdot 3)}{3 x - 9}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $3$ .

$\frac{\sin (3 - 3)}{3 x - 9}$

Step 3.1.2

Subtract $3$ from $3$ .

$\frac{\sin (0)}{3 x - 9}$

Step 3.1.3

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

$\frac{0}{3 x - 9}$

Step 3.2

Factor $3$ out of $3 x - 9$ .

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Step 3.2.1

Factor $3$ out of $3 x$ .

$\frac{0}{3 (x) - 9}$

Step 3.2.2

Factor $3$ out of $- 9$ .

$\frac{0}{3 x + 3 \cdot - 3}$

Step 3.2.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$\frac{0}{3 (x - 3)}$

Step 3.3

Cancel the common factor of $0$ and $3$ .

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Step 3.3.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3 (x - 3)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

$\frac{0}{x - 3}$

Step 3.4

Divide $0$ by $x - 3$ .

$0$