Calculus Examples

$lim_{x \to - 8} \frac{80}{e^{x} - 1}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $80$ outside of the limit because it is constant with respect to $x$ .

$80 lim_{x \to - 8} \frac{1}{e^{x} - 1}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$80 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} e^{x} - 1}$

Step 1.3

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

$80 \frac{1}{lim_{x \to - 8} e^{x} - 1}$

Step 1.4

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

$80 \frac{1}{lim_{x \to - 8} e^{x} - lim_{x \to - 8} 1}$

Step 1.5

Move the limit into the exponent.

$80 \frac{1}{e^{lim_{x \to - 8} x} - lim_{x \to - 8} 1}$

Step 1.6

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

$80 \frac{1}{e^{lim_{x \to - 8} x} - 1 \cdot 1}$

Step 2

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

$80 \frac{1}{e^{- 8} - 1 \cdot 1}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Rewrite $e^{- 8}$ as ${(e^{- 4})}^{2}$ .

$80 \frac{1}{{(e^{- 4})}^{2} - 1 \cdot 1}$

Step 3.1.2

Rewrite $1$ as $1^{2}$ .

$80 \frac{1}{{(e^{- 4})}^{2} - 1^{2}}$

Step 3.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{- 4}$ and $b = 1$ .

$80 \frac{1}{(e^{- 4} + 1) (e^{- 4} - 1)}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

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

$80 \frac{1}{(\frac{1}{e^{4}} + 1) (e^{- 4} - 1)}$

Step 3.1.4.2

Rewrite $e^{- 4}$ as ${(e^{- 2})}^{2}$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ({(e^{- 2})}^{2} - 1)}$

Step 3.1.4.3

Rewrite $1$ as $1^{2}$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ({(e^{- 2})}^{2} - 1^{2})}$

Step 3.1.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{- 2}$ and $b = 1$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((e^{- 2} + 1) (e^{- 2} - 1))}$

Step 3.1.4.5

Simplify.

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Step 3.1.4.5.1

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

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) (e^{- 2} - 1))}$

Step 3.1.4.5.2

Rewrite $e^{- 2}$ as ${(e^{- 1})}^{2}$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) ({(e^{- 1})}^{2} - 1))}$

Step 3.1.4.5.3

Rewrite $1$ as $1^{2}$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) ({(e^{- 1})}^{2} - 1^{2}))}$

Step 3.1.4.5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e^{- 1}$ and $b = 1$ .

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) ((e^{- 1} + 1) (e^{- 1} - 1)))}$

Step 3.1.4.5.5

Simplify.

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Step 3.1.4.5.5.1

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

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) ((\frac{1}{e} + 1) (e^{- 1} - 1)))}$

Step 3.1.4.5.5.2

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

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) ((\frac{1}{e} + 1) (\frac{1}{e} - 1)))}$

$80 \frac{1}{(\frac{1}{e^{4}} + 1) ((\frac{1}{e^{2}} + 1) (\frac{1}{e} + 1) (\frac{1}{e} - 1))}$

$80 \frac{1}{(\frac{1}{e^{4}} + 1) (\frac{1}{e^{2}} + 1) (\frac{1}{e} + 1) (\frac{1}{e} - 1)}$

Step 3.1.5

Write $1$ as a fraction with a common denominator.

$80 \frac{1}{(\frac{1}{e^{4}} + \frac{e^{4}}{e^{4}}) (\frac{1}{e^{2}} + 1) (\frac{1}{e} + 1) (\frac{1}{e} - 1)}$

Step 3.1.6

Combine the numerators over the common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} (\frac{1}{e^{2}} + 1) (\frac{1}{e} + 1) (\frac{1}{e} - 1)}$

Step 3.1.7

Write $1$ as a fraction with a common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} (\frac{1}{e^{2}} + \frac{e^{2}}{e^{2}}) (\frac{1}{e} + 1) (\frac{1}{e} - 1)}$

Step 3.1.8

Combine the numerators over the common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} (\frac{1}{e} + 1) (\frac{1}{e} - 1)}$

Step 3.1.9

Write $1$ as a fraction with a common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} (\frac{1}{e} + \frac{e}{e}) (\frac{1}{e} - 1)}$

Step 3.1.10

Combine the numerators over the common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} \frac{1 + e}{e} (\frac{1}{e} - 1)}$

Step 3.1.11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} \frac{1 + e}{e} (\frac{1}{e} - 1 \cdot \frac{e}{e})}$

Step 3.1.12

Combine $- 1$ and $\frac{e}{e}$ .

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} \frac{1 + e}{e} (\frac{1}{e} + \frac{- e}{e})}$

Step 3.1.13

Combine the numerators over the common denominator.

$80 \frac{1}{\frac{1 + e^{4}}{e^{4}} \cdot \frac{1 + e^{2}}{e^{2}} \frac{1 + e}{e} \frac{1 - e}{e}}$

Step 3.1.14

Combine exponents.

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Step 3.1.14.1

Multiply $\frac{1 + e^{4}}{e^{4}}$ by $\frac{1 + e^{2}}{e^{2}}$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2})}{e^{4} e^{2}} \cdot \frac{1 + e}{e} \frac{1 - e}{e}}$

Step 3.1.14.2

Multiply $e^{4}$ by $e^{2}$ by adding the exponents.

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Step 3.1.14.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2})}{e^{4 + 2}} \cdot \frac{1 + e}{e} \frac{1 - e}{e}}$

Step 3.1.14.2.2

Add $4$ and $2$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2})}{e^{6}} \cdot \frac{1 + e}{e} \frac{1 - e}{e}}$

Step 3.1.14.3

Multiply $\frac{(1 + e^{4}) (1 + e^{2})}{e^{6}}$ by $\frac{1 + e}{e}$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e)}{e^{6} e} \cdot \frac{1 - e}{e}}$

Step 3.1.14.4

Multiply $e^{6}$ by $e$ by adding the exponents.

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Step 3.1.14.4.1

Multiply $e^{6}$ by $e$ .

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Step 3.1.14.4.1.1

Raise $e$ to the power of $1$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e)}{e^{6} e^{1}} \cdot \frac{1 - e}{e}}$

Step 3.1.14.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e)}{e^{6 + 1}} \cdot \frac{1 - e}{e}}$

Step 3.1.14.4.2

Add $6$ and $1$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e)}{e^{7}} \cdot \frac{1 - e}{e}}$

Step 3.1.14.5

Multiply $\frac{(1 + e^{4}) (1 + e^{2}) (1 + e)}{e^{7}}$ by $\frac{1 - e}{e}$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{e^{7} e}}$

Step 3.1.14.6

Multiply $e^{7}$ by $e$ by adding the exponents.

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Step 3.1.14.6.1

Multiply $e^{7}$ by $e$ .

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Step 3.1.14.6.1.1

Raise $e$ to the power of $1$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{e^{7} e^{1}}}$

Step 3.1.14.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{e^{7 + 1}}}$

Step 3.1.14.6.2

Add $7$ and $1$ .

$80 \frac{1}{\frac{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}{e^{8}}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$80 (1 \frac{e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)})$

Step 3.3

Multiply $\frac{e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$ by $1$ .

$80 \frac{e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$

Step 3.4

Combine $80$ and $\frac{e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$ .

$\frac{80 e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{80 e^{8}}{(1 + e^{4}) (1 + e^{2}) (1 + e) (1 - e)}$

Decimal Form:

$- 80.02684601 \dots$