Calculus Examples

$\frac{lim_{x \to 3} (x^{2} + 2 x - 15) \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 1

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

$\frac{lim_{x \to 3} x^{2} + 2 x - 15 \cdot lim_{x \to 3} \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 2

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

$\frac{(lim_{x \to 3} x^{2} + lim_{x \to 3} 2 x - lim_{x \to 3} 15) \cdot lim_{x \to 3} \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 3

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

$\frac{({(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 2 x - lim_{x \to 3} 15) \cdot lim_{x \to 3} \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 4

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

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - lim_{x \to 3} 15) \cdot lim_{x \to 3} \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 5

Evaluate the limit of $15$ which is constant as $x$ approaches $3$ .

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot lim_{x \to 3} \sin (9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 6

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

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot \sin (lim_{x \to 3} 9 - 3 x)}{\tan^{2} (2 x - 6)}$

Step 7

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

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot \sin (lim_{x \to 3} 9 - lim_{x \to 3} 3 x)}{\tan^{2} (2 x - 6)}$

Step 8

Evaluate the limit of $9$ which is constant as $x$ approaches $3$ .

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot \sin (9 - lim_{x \to 3} 3 x)}{\tan^{2} (2 x - 6)}$

Step 9

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{({(lim_{x \to 3} x)}^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot \sin (9 - 1 \cdot 3 lim_{x \to 3} x)}{\tan^{2} (2 x - 6)}$

Step 10

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 10.1

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

$\frac{(3^{2} + 2 lim_{x \to 3} x - 1 \cdot 15) \cdot \sin (9 - 1 \cdot 3 lim_{x \to 3} x)}{\tan^{2} (2 x - 6)}$

Step 10.2

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

$\frac{(3^{2} + 2 \cdot 3 - 1 \cdot 15) \cdot \sin (9 - 1 \cdot 3 lim_{x \to 3} x)}{\tan^{2} (2 x - 6)}$

Step 10.3

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

$\frac{(3^{2} + 2 \cdot 3 - 1 \cdot 15) \cdot \sin (9 - 1 \cdot 3 \cdot 3)}{\tan^{2} (2 x - 6)}$

Step 11

Simplify the answer.

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Step 11.1

Factor $\tan (2 x - 6)$ out of $\tan^{2} (2 x - 6)$ .

$\frac{(3^{2} + 2 \cdot 3 - 1 \cdot 15) \cdot \sin (9 - 1 \cdot 3 \cdot 3)}{\tan (2 x - 6) \tan (2 x - 6)}$

Step 11.2

Separate fractions.

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3)}{\tan (2 x - 6)}$

Step 11.3

Rewrite $\tan (2 x - 6)$ in terms of sines and cosines.

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3)}{\frac{\sin (2 x - 6)}{\cos (2 x - 6)}}$

Step 11.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (2 x - 6)}{\cos (2 x - 6)}$ .

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} (\sin (9 - 1 \cdot 3 \cdot 3) \frac{\cos (2 x - 6)}{\sin (2 x - 6)})$

Step 11.5

Write $\sin (9 - 1 \cdot 3 \cdot 3)$ as a fraction with denominator $1$ .

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} (\frac{\sin (9 - 1 \cdot 3 \cdot 3)}{1} \cdot \frac{\cos (2 x - 6)}{\sin (2 x - 6)})$

Step 11.6

Simplify.

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Step 11.6.1

Divide $\sin (9 - 1 \cdot 3 \cdot 3)$ by $1$ .

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} (\sin (9 - 1 \cdot 3 \cdot 3) \frac{\cos (2 x - 6)}{\sin (2 x - 6)})$

Step 11.6.2

Combine $\sin (9 - 1 \cdot 3 \cdot 3)$ and $\frac{\cos (2 x - 6)}{\sin (2 x - 6)}$ .

$\frac{3^{2} + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.7

Simplify the numerator.

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Step 11.7.1

Raise $3$ to the power of $2$ .

$\frac{9 + 2 \cdot 3 - 1 \cdot 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.7.2

Multiply $2$ by $3$ .

$\frac{9 + 6 - 1 \cdot 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.7.3

Multiply $- 1$ by $15$ .

$\frac{9 + 6 - 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.7.4

Add $9$ and $6$ .

$\frac{15 - 15}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.7.5

Subtract $15$ from $15$ .

$\frac{0}{\tan (2 x - 6)} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.8

Rewrite $\tan (2 x - 6)$ in terms of sines and cosines.

$\frac{0}{\frac{\sin (2 x - 6)}{\cos (2 x - 6)}} \cdot \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.9

Multiply the numerator by the reciprocal of the denominator.

$0 \frac{\cos (2 x - 6)}{\sin (2 x - 6)} \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.10

Multiply $0$ by $\frac{\cos (2 x - 6)}{\sin (2 x - 6)}$ .

$0 \frac{\sin (9 - 1 \cdot 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.11

Simplify the numerator.

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Step 11.11.1

Multiply $- 1 \cdot 3 \cdot 3$ .

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Step 11.11.1.1

Multiply $- 1$ by $3$ .

$0 \frac{\sin (9 - 3 \cdot 3) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.11.1.2

Multiply $- 3$ by $3$ .

$0 \frac{\sin (9 - 9) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.11.2

Subtract $9$ from $9$ .

$0 \frac{\sin (0) \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.11.3

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

$0 \frac{0 \cos (2 x - 6)}{\sin (2 x - 6)}$

Step 11.12

Multiply $0$ by $\cos (2 x - 6)$ .

$0 \frac{0}{\sin (2 x - 6)}$

Step 11.13

Divide $0$ by $\sin (2 x - 6)$ .

$0 \cdot 0$

Step 11.14

Multiply $0$ by $0$ .

$0$