Calculus Examples

$lim_{x \to 8} 20000 - 20000 e^{- 0.1 x}$

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 $x$ approaches $8$ .

$lim_{x \to 8} 20000 - lim_{x \to 8} 20000 e^{- 0.1 x}$

Step 1.2

Evaluate the limit of $20000$ which is constant as $x$ approaches $8$ .

$20000 - lim_{x \to 8} 20000 e^{- 0.1 x}$

Step 1.3

Move the term $20000$ outside of the limit because it is constant with respect to $x$ .

$20000 - 20000 lim_{x \to 8} e^{- 0.1 x}$

Step 1.4

Move the limit into the exponent.

$20000 - 20000 e^{lim_{x \to 8} - 0.1 x}$

Step 1.5

Move the term $- 0.1$ outside of the limit because it is constant with respect to $x$ .

$20000 - 20000 e^{- 0.1 lim_{x \to 8} x}$

Step 2

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

$20000 - 20000 e^{- 0.1 \cdot 8}$

Step 3

Simplify each term.

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Step 3.1

Multiply $- 0.1$ by $8$ .

$20000 - 20000 e^{- 0.8}$

Step 3.2

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

$20000 - 20000 \frac{1}{e^{0.8}}$

Step 3.3

Combine $- 20000$ and $\frac{1}{e^{0.8}}$ .

$20000 + \frac{- 20000}{e^{0.8}}$

Step 3.4

Move the negative in front of the fraction.

$20000 - \frac{20000}{e^{0.8}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$20000 - \frac{20000}{e^{0.8}}$

Decimal Form:

$11013.42071765 \dots$