Calculus Examples

$\frac{lim_{x \to 3} \sin (2 x - 6)}{x^{2} - 9}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 3} 2 x - 6)}{x^{2} - 9}$

Step 1.2

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

$\frac{\sin (lim_{x \to 3} 2 x - lim_{x \to 3} 6)}{x^{2} - 9}$

Step 1.3

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

$\frac{\sin (2 lim_{x \to 3} x - lim_{x \to 3} 6)}{x^{2} - 9}$

Step 1.4

Evaluate the limit of $6$ which is constant as $x$ approaches $3$ .

$\frac{\sin (2 lim_{x \to 3} x - 1 \cdot 6)}{x^{2} - 9}$

Step 2

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

$\frac{\sin (2 \cdot 3 - 1 \cdot 6)}{x^{2} - 9}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply $2$ by $3$ .

$\frac{\sin (6 - 1 \cdot 6)}{x^{2} - 9}$

Step 3.1.1.2

Multiply $- 1$ by $6$ .

$\frac{\sin (6 - 6)}{x^{2} - 9}$

Step 3.1.2

Subtract $6$ from $6$ .

$\frac{\sin (0)}{x^{2} - 9}$

Step 3.1.3

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

$\frac{0}{x^{2} - 9}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{0}{x^{2} - 3^{2}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 3.3

Divide $0$ by $(x + 3) (x - 3)$ .

$0$