Calculus Examples

$lim_{t \to 8} \frac{- e^{t} \cos (t) + e^{t} \sin (t)}{2}$

Step 1

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

$\frac{1}{2} lim_{t \to 8} - e^{t} \cos (t) + e^{t} \sin (t)$

Step 2

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

$\frac{1}{2} (- lim_{t \to 8} e^{t} \cos (t) + lim_{t \to 8} e^{t} \sin (t))$

Step 3

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

$\frac{1}{2} (- lim_{t \to 8} e^{t} \cdot lim_{t \to 8} \cos (t) + lim_{t \to 8} e^{t} \sin (t))$

Step 4

Move the limit into the exponent.

$\frac{1}{2} (- e^{lim_{t \to 8} t} \cdot lim_{t \to 8} \cos (t) + lim_{t \to 8} e^{t} \sin (t))$

Step 5

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

$\frac{1}{2} (- e^{lim_{t \to 8} t} \cdot \cos (lim_{t \to 8} t) + lim_{t \to 8} e^{t} \sin (t))$

Step 6

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

$\frac{1}{2} (- e^{lim_{t \to 8} t} \cdot \cos (lim_{t \to 8} t) + lim_{t \to 8} e^{t} \cdot lim_{t \to 8} \sin (t))$

Step 7

Move the limit into the exponent.

$\frac{1}{2} (- e^{lim_{t \to 8} t} \cdot \cos (lim_{t \to 8} t) + e^{lim_{t \to 8} t} \cdot lim_{t \to 8} \sin (t))$

Step 8

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

$\frac{1}{2} (- e^{lim_{t \to 8} t} \cdot \cos (lim_{t \to 8} t) + e^{lim_{t \to 8} t} \cdot \sin (lim_{t \to 8} t))$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $t$ .

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Step 9.1

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

$\frac{1}{2} (- e^{8} \cdot \cos (lim_{t \to 8} t) + e^{lim_{t \to 8} t} \cdot \sin (lim_{t \to 8} t))$

Step 9.2

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

$\frac{1}{2} (- e^{8} \cdot \cos (8) + e^{lim_{t \to 8} t} \cdot \sin (lim_{t \to 8} t))$

Step 9.3

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

$\frac{1}{2} (- e^{8} \cdot \cos (8) + e^{8} \cdot \sin (lim_{t \to 8} t))$

Step 9.4

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

$\frac{1}{2} (- e^{8} \cdot \cos (8) + e^{8} \cdot \sin (8))$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Evaluate $\cos (8)$ .

$\frac{1}{2} (- e^{8} \cdot 0.99026806 + e^{8} \cdot \sin (8))$

Step 10.1.2

Multiply $- e^{8} \cdot 0.99026806$ .

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Step 10.1.2.1

Multiply $0.99026806$ by $- 1$ .

$\frac{1}{2} (- 0.99026806 e^{8} + e^{8} \cdot \sin (8))$

Step 10.1.2.2

Multiply $- 0.99026806$ by $e^{8}$ .

$\frac{1}{2} (- 2951.94750882 + e^{8} \cdot \sin (8))$

Step 10.1.3

Evaluate $\sin (8)$ .

$\frac{1}{2} (- 2951.94750882 + e^{8} \cdot 0.1391731)$

Step 10.1.4

Multiply $e^{8}$ by $0.1391731$ .

$\frac{1}{2} (- 2951.94750882 + 414.86916688)$

Step 10.2

Add $- 2951.94750882$ and $414.86916688$ .

$\frac{1}{2} \cdot - 2537.07834193$

Step 10.3

Combine $\frac{1}{2}$ and $- 2537.07834193$ .

$\frac{- 2537.07834193}{2}$

Step 10.4

Divide $- 2537.07834193$ by $2$ .

$- 1268.53917096$