Calculus Examples

$lim_{x \to 3} - 9 π \cos (\frac{π x}{2}) \sin (\frac{π x}{2})$

Step 1

Move the term $- 9 π$ outside of the limit because it is constant with respect to $x$ .

$- 9 π lim_{x \to 3} \cos (\frac{π x}{2}) \sin (\frac{π x}{2})$

Step 2

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

$- 9 π lim_{x \to 3} \cos (\frac{π x}{2}) \cdot lim_{x \to 3} \sin (\frac{π x}{2})$

Step 3

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

$- 9 π \cos (lim_{x \to 3} \frac{π x}{2}) \cdot lim_{x \to 3} \sin (\frac{π x}{2})$

Step 4

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$- 9 π \cos (\frac{π}{2} lim_{x \to 3} x) \cdot lim_{x \to 3} \sin (\frac{π x}{2})$

Step 5

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

$- 9 π \cos (\frac{π}{2} lim_{x \to 3} x) \cdot \sin (lim_{x \to 3} \frac{π x}{2})$

Step 6

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$- 9 π \cos (\frac{π}{2} lim_{x \to 3} x) \cdot \sin (\frac{π}{2} lim_{x \to 3} x)$

Step 7

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

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Step 7.1

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

$- 9 π \cos (\frac{π}{2} \cdot 3) \cdot \sin (\frac{π}{2} lim_{x \to 3} x)$

Step 7.2

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

$- 9 π \cos (\frac{π}{2} \cdot 3) \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{π}{2}$ and $3$ .

$- 9 π \cos (\frac{π \cdot 3}{2}) \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.2

Move $3$ to the left of $π$ .

$- 9 π \cos (\frac{3 \cdot π}{2}) \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 9 π \cos (\frac{π}{2}) \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- 9 π \cdot 0 \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.5

Multiply $- 9 π \cdot 0$ .

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Step 8.5.1

Multiply $0$ by $- 9$ .

$0 π \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.5.2

Multiply $0$ by $π$ .

$0 \cdot \sin (\frac{π}{2} \cdot 3)$

Step 8.6

Combine $\frac{π}{2}$ and $3$ .

$0 \cdot \sin (\frac{π \cdot 3}{2})$

Step 8.7

Move $3$ to the left of $π$ .

$0 \cdot \sin (\frac{3 \cdot π}{2})$

Step 8.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$0 \cdot (- \sin (\frac{π}{2}))$

Step 8.9

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 \cdot (- 1 \cdot 1)$

Step 8.10

Multiply $0 (- 1 \cdot 1)$ .

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Step 8.10.1

Multiply $- 1$ by $1$ .

$0 \cdot - 1$

Step 8.10.2

Multiply $0$ by $- 1$ .

$0$