Calculus Examples

$lim_{t \to 8} (12 + 2.8) \cdot \sin (\frac{2 π}{365} \cdot (t - 80))$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $12 + 2.8$ outside of the limit because it is constant with respect to $t$ .

$(12 + 2.8) lim_{t \to 8} \sin (\frac{2 π}{365} \cdot (t - 80))$

Step 1.2

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

$(12 + 2.8) \sin (lim_{t \to 8} \frac{2 π}{365} \cdot (t - 80))$

Step 1.3

Move the term $\frac{2 π}{365}$ outside of the limit because it is constant with respect to $t$ .

$(12 + 2.8) \sin (\frac{2 π}{365} lim_{t \to 8} t - 80)$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $8$ .

$(12 + 2.8) \sin (\frac{2 π}{365} (lim_{t \to 8} t - lim_{t \to 8} 80))$

Step 1.5

Evaluate the limit of $80$ which is constant as $t$ approaches $8$ .

$(12 + 2.8) \sin (\frac{2 π}{365} (lim_{t \to 8} t - 1 \cdot 80))$

Step 2

Evaluate the limit of $t$ by plugging in $8$ for $t$ .

$(12 + 2.8) \sin (\frac{2 π}{365} (8 - 1 \cdot 80))$

Step 3

Simplify the answer.

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Step 3.1

Add $12$ and $2.8$ .

$14.8 \sin (\frac{2 π}{365} (8 - 1 \cdot 80))$

Step 3.2

Multiply $- 1$ by $80$ .

$14.8 \sin (\frac{2 π}{365} (8 - 80))$

Step 3.3

Subtract $80$ from $8$ .

$14.8 \sin (\frac{2 π}{365} \cdot - 72)$

Step 3.4

Multiply $\frac{2 π}{365} \cdot - 72$ .

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Step 3.4.1

Combine $\frac{2 π}{365}$ and $- 72$ .

$14.8 \sin (\frac{2 π \cdot - 72}{365})$

Step 3.4.2

Multiply $- 72$ by $2$ .

$14.8 \sin (\frac{- 144 π}{365})$

Step 3.5

Move the negative in front of the fraction.

$14.8 \sin (- \frac{144 π}{365})$

Step 3.6

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$14.8 \sin (\frac{586 π}{365})$

Step 3.7

Evaluate $\sin (\frac{586 π}{365})$ .

$14.8 \cdot - 0.94559638$

Step 3.8

Multiply $14.8$ by $- 0.94559638$ .

$- 13.99482653$