Calculus Examples

$lim_{t \to 0} - 2 \sin (\frac{m t + n t}{2}) \sin (\frac{m t - n t}{2})$

Step 1

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

$- 2 lim_{t \to 0} \sin (\frac{m t + n t}{2}) \sin (\frac{m t - n t}{2})$

Step 2

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

$- 2 lim_{t \to 0} \sin (\frac{m t + n t}{2}) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 3

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

$- 2 \sin (lim_{t \to 0} \frac{m t + n t}{2}) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} lim_{t \to 0} m t + n t) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$- 2 \sin (\frac{1}{2} (lim_{t \to 0} m t + lim_{t \to 0} n t)) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 6

Move the term $m$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + lim_{t \to 0} n t)) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 7

Move the term $n$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot lim_{t \to 0} \sin (\frac{m t - n t}{2})$

Step 8

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

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot \sin (lim_{t \to 0} \frac{m t - n t}{2})$

Step 9

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot \sin (\frac{1}{2} lim_{t \to 0} m t - n t)$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot \sin (\frac{1}{2} (lim_{t \to 0} m t - lim_{t \to 0} n t))$

Step 11

Move the term $m$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot \sin (\frac{1}{2} (m lim_{t \to 0} t - lim_{t \to 0} n t))$

Step 12

Move the term $n$ outside of the limit because it is constant with respect to $t$ .

$- 2 \sin (\frac{1}{2} (m lim_{t \to 0} t + n lim_{t \to 0} t)) \cdot \sin (\frac{1}{2} (m lim_{t \to 0} t - n lim_{t \to 0} t))$

Step 13

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

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Step 13.1

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

$- 2 \sin (\frac{1}{2} (m \cdot 0 + n lim_{t \to 0} t)) \cdot \sin (\frac{1}{2} (m lim_{t \to 0} t - n lim_{t \to 0} t))$

Step 13.2

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

$- 2 \sin (\frac{1}{2} (m \cdot 0 + n \cdot 0)) \cdot \sin (\frac{1}{2} (m lim_{t \to 0} t - n lim_{t \to 0} t))$

Step 13.3

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

$- 2 \sin (\frac{1}{2} (m \cdot 0 + n \cdot 0)) \cdot \sin (\frac{1}{2} (m \cdot 0 - n lim_{t \to 0} t))$

Step 13.4

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

$- 2 \sin (\frac{1}{2} (m \cdot 0 + n \cdot 0)) \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14

Simplify the answer.

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Step 14.1

Simplify each term.

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Step 14.1.1

Multiply $m$ by $0$ .

$- 2 \sin (\frac{1}{2} (0 + n \cdot 0)) \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.1.2

Multiply $n$ by $0$ .

$- 2 \sin (\frac{1}{2} (0 + 0)) \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.2

Add $0$ and $0$ .

$- 2 \sin (\frac{1}{2} \cdot 0) \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.3

Multiply $\frac{1}{2}$ by $0$ .

$- 2 \sin (0) \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.4

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

$- 2 \cdot 0 \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.5

Multiply $- 2$ by $0$ .

$0 \cdot \sin (\frac{1}{2} (m \cdot 0 - n \cdot 0))$

Step 14.6

Simplify each term.

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Step 14.6.1

Multiply $m$ by $0$ .

$0 \cdot \sin (\frac{1}{2} (0 - n \cdot 0))$

Step 14.6.2

Multiply $- n \cdot 0$ .

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Step 14.6.2.1

Multiply $0$ by $- 1$ .

$0 \cdot \sin (\frac{1}{2} (0 + 0 n))$

Step 14.6.2.2

Multiply $0$ by $n$ .

$0 \cdot \sin (\frac{1}{2} (0 + 0))$

Step 14.7

Add $0$ and $0$ .

$0 \cdot \sin (\frac{1}{2} \cdot 0)$

Step 14.8

Multiply $\frac{1}{2}$ by $0$ .

$0 \cdot \sin (0)$

Step 14.9

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

$0 \cdot 0$

Step 14.10

Multiply $0$ by $0$ .

$0$