Calculus Examples

$lim_{t \to 8} 283 + 3 (1 - e^{- 0.03 t})$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{t \to 8} 283 + lim_{t \to 8} 3 (1 - e^{- 0.03 t})$

Step 1.2

Evaluate the limit of $283$ which is constant as $t$ approaches $8$ .

$283 + lim_{t \to 8} 3 (1 - e^{- 0.03 t})$

Step 1.3

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

$283 + 3 lim_{t \to 8} 1 - e^{- 0.03 t}$

Step 1.4

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

$283 + 3 (lim_{t \to 8} 1 - lim_{t \to 8} e^{- 0.03 t})$

Step 1.5

Evaluate the limit of $1$ which is constant as $t$ approaches $8$ .

$283 + 3 (1 - lim_{t \to 8} e^{- 0.03 t})$

Step 1.6

Move the limit into the exponent.

$283 + 3 (1 - e^{lim_{t \to 8} - 0.03 t})$

Step 1.7

Move the term $- 0.03$ outside of the limit because it is constant with respect to $t$ .

$283 + 3 (1 - e^{- 0.03 lim_{t \to 8} t})$

Step 2

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

$283 + 3 (1 - e^{- 0.03 \cdot 8})$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply $- 0.03$ by $8$ .

$283 + 3 (1 - e^{- 0.24})$

Step 3.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$283 + 3 (1 - \frac{1}{e^{0.24}})$

Step 3.1.2

Apply the distributive property.

$283 + 3 \cdot 1 + 3 (- \frac{1}{e^{0.24}})$

Step 3.1.3

Multiply $3$ by $1$ .

$283 + 3 + 3 (- \frac{1}{e^{0.24}})$

Step 3.1.4

Multiply $3 (- \frac{1}{e^{0.24}})$ .

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Step 3.1.4.1

Multiply $- 1$ by $3$ .

$283 + 3 - 3 \frac{1}{e^{0.24}}$

Step 3.1.4.2

Combine $- 3$ and $\frac{1}{e^{0.24}}$ .

$283 + 3 + \frac{- 3}{e^{0.24}}$

Step 3.1.5

Move the negative in front of the fraction.

$283 + 3 - \frac{3}{e^{0.24}}$

Step 3.2

Add $283$ and $3$ .

$286 - \frac{3}{e^{0.24}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$286 - \frac{3}{e^{0.24}}$

Decimal Form:

$283.64011641 \dots$