Calculus Examples

$lim_{x \to - 8} \frac{4 e^{x} - e^{- x}}{e^{x} - 5 e^{- x}}$

Step 1

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

$\frac{lim_{x \to - 8} 4 e^{x} - e^{- x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 2

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

$\frac{lim_{x \to - 8} 4 e^{x} - lim_{x \to - 8} e^{- x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to - 8} e^{x} - lim_{x \to - 8} e^{- x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 4

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to - 8} x} - lim_{x \to - 8} e^{- x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 5

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to - 8} x} - e^{lim_{x \to - 8} - x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 6

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{lim_{x \to - 8} e^{x} - 5 e^{- x}}$

Step 7

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

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{lim_{x \to - 8} e^{x} - lim_{x \to - 8} 5 e^{- x}}$

Step 8

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{e^{lim_{x \to - 8} x} - lim_{x \to - 8} 5 e^{- x}}$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{e^{lim_{x \to - 8} x} - 5 lim_{x \to - 8} e^{- x}}$

Step 10

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{e^{lim_{x \to - 8} x} - 5 e^{lim_{x \to - 8} - x}}$

Step 11

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 e^{lim_{x \to - 8} x} - e^{- lim_{x \to - 8} x}}{e^{lim_{x \to - 8} x} - 5 e^{- lim_{x \to - 8} x}}$

Step 12

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

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Step 12.1

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

$\frac{4 e^{- 8} - e^{- lim_{x \to - 8} x}}{e^{lim_{x \to - 8} x} - 5 e^{- lim_{x \to - 8} x}}$

Step 12.2

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

$\frac{4 e^{- 8} - e^{8}}{e^{lim_{x \to - 8} x} - 5 e^{- lim_{x \to - 8} x}}$

Step 12.3

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

$\frac{4 e^{- 8} - e^{8}}{e^{- 8} - 5 e^{- lim_{x \to - 8} x}}$

Step 12.4

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

$\frac{4 e^{- 8} - e^{8}}{e^{- 8} - 5 e^{8}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{{(2 e^{- 4})}^{2} - e^{8}}{e^{- 8} - 5 e^{8}}$

Step 13.1.2

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

$\frac{{(2 e^{- 4})}^{2} - {(e^{4})}^{2}}{e^{- 8} - 5 e^{8}}$

Step 13.1.3

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

$\frac{(2 e^{- 4} + e^{4}) (2 e^{- 4} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.4

Simplify.

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Step 13.1.4.1

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

$\frac{(2 \frac{1}{e^{4}} + e^{4}) (2 e^{- 4} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.4.2

Combine $2$ and $\frac{1}{e^{4}}$ .

$\frac{(\frac{2}{e^{4}} + e^{4}) (2 e^{- 4} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.4.3

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

$\frac{(\frac{2}{e^{4}} + e^{4}) (2 \frac{1}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.4.4

Combine $2$ and $\frac{1}{e^{4}}$ .

$\frac{(\frac{2}{e^{4}} + e^{4}) (\frac{2}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.5

To write $e^{4}$ as a fraction with a common denominator, multiply by $\frac{e^{4}}{e^{4}}$ .

$\frac{(\frac{2}{e^{4}} + \frac{e^{4} e^{4}}{e^{4}}) (\frac{2}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 + e^{4} e^{4}}{e^{4}} (\frac{2}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.7

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

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Step 13.1.7.1

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

$\frac{\frac{2 + e^{4 + 4}}{e^{4}} (\frac{2}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.7.2

Add $4$ and $4$ .

$\frac{\frac{2 + e^{8}}{e^{4}} (\frac{2}{e^{4}} - e^{4})}{e^{- 8} - 5 e^{8}}$

Step 13.1.8

To write $- e^{4}$ as a fraction with a common denominator, multiply by $\frac{e^{4}}{e^{4}}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} (\frac{2}{e^{4}} - e^{4} \cdot \frac{e^{4}}{e^{4}})}{e^{- 8} - 5 e^{8}}$

Step 13.1.9

Combine $- e^{4}$ and $\frac{e^{4}}{e^{4}}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} (\frac{2}{e^{4}} + \frac{- e^{4} e^{4}}{e^{4}})}{e^{- 8} - 5 e^{8}}$

Step 13.1.10

Combine the numerators over the common denominator.

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{4} e^{4}}{e^{4}}}{e^{- 8} - 5 e^{8}}$

Step 13.1.11

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

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Step 13.1.11.1

Move $e^{4}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - (e^{4} e^{4})}{e^{4}}}{e^{- 8} - 5 e^{8}}$

Step 13.1.11.2

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

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{4 + 4}}{e^{4}}}{e^{- 8} - 5 e^{8}}$

Step 13.1.11.3

Add $4$ and $4$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{e^{- 8} - 5 e^{8}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1}{e^{8}} - 5 e^{8}}$

Step 13.2.2

To write $- 5 e^{8}$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1}{e^{8}} - 5 e^{8} \cdot \frac{e^{8}}{e^{8}}}$

Step 13.2.3

Combine $- 5 e^{8}$ and $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1}{e^{8}} + \frac{- 5 e^{8} e^{8}}{e^{8}}}$

Step 13.2.4

Combine the numerators over the common denominator.

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1 - 5 e^{8} e^{8}}{e^{8}}}$

Step 13.2.5

Multiply $e^{8}$ by $e^{8}$ by adding the exponents.

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Step 13.2.5.1

Move $e^{8}$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1 - 5 (e^{8} e^{8})}{e^{8}}}$

Step 13.2.5.2

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

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1 - 5 e^{8 + 8}}{e^{8}}}$

Step 13.2.5.3

Add $8$ and $8$ .

$\frac{\frac{2 + e^{8}}{e^{4}} \cdot \frac{2 - e^{8}}{e^{4}}}{\frac{1 - 5 e^{16}}{e^{8}}}$

Step 13.3

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

$\frac{\frac{(2 + e^{8}) (2 - e^{8})}{e^{4} e^{4}}}{\frac{1 - 5 e^{16}}{e^{8}}}$

Step 13.4

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

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Step 13.4.1

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

$\frac{\frac{(2 + e^{8}) (2 - e^{8})}{e^{4 + 4}}}{\frac{1 - 5 e^{16}}{e^{8}}}$

Step 13.4.2

Add $4$ and $4$ .

$\frac{\frac{(2 + e^{8}) (2 - e^{8})}{e^{8}}}{\frac{1 - 5 e^{16}}{e^{8}}}$

Step 13.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{(2 + e^{8}) (2 - e^{8})}{e^{8}} \cdot \frac{e^{8}}{1 - 5 e^{16}}$

Step 13.6

Cancel the common factor of $e^{8}$ .

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Step 13.6.1

Cancel the common factor.

$\frac{(2 + e^{8}) (2 - e^{8})}{e^{8}} \cdot \frac{e^{8}}{1 - 5 e^{16}}$

Step 13.6.2

Rewrite the expression.

$(2 + e^{8}) (2 - e^{8}) \frac{1}{1 - 5 e^{16}}$

Step 13.7

Expand $(2 + e^{8}) (2 - e^{8})$ using the FOIL Method.

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Step 13.7.1

Apply the distributive property.

$(2 (2 - e^{8}) + e^{8} (2 - e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.7.2

Apply the distributive property.

$(2 \cdot 2 + 2 (- e^{8}) + e^{8} (2 - e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.7.3

Apply the distributive property.

$(2 \cdot 2 + 2 (- e^{8}) + e^{8} \cdot 2 + e^{8} (- e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.8

Simplify and combine like terms.

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Step 13.8.1

Simplify each term.

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Step 13.8.1.1

Multiply $2$ by $2$ .

$(4 + 2 (- e^{8}) + e^{8} \cdot 2 + e^{8} (- e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.2

Multiply $- 1$ by $2$ .

$(4 - 2 e^{8} + e^{8} \cdot 2 + e^{8} (- e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.3

Move $2$ to the left of $e^{8}$ .

$(4 - 2 e^{8} + 2 \cdot e^{8} + e^{8} (- e^{8})) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.4

Multiply $e^{8}$ by $e^{8}$ by adding the exponents.

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Step 13.8.1.4.1

Move $e^{8}$ .

$(4 - 2 e^{8} + 2 e^{8} + e^{8} e^{8} \cdot - 1) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.4.2

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

$(4 - 2 e^{8} + 2 e^{8} + e^{8 + 8} \cdot - 1) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.4.3

Add $8$ and $8$ .

$(4 - 2 e^{8} + 2 e^{8} + e^{16} \cdot - 1) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.5

Move $- 1$ to the left of $e^{16}$ .

$(4 - 2 e^{8} + 2 e^{8} - 1 \cdot e^{16}) \frac{1}{1 - 5 e^{16}}$

Step 13.8.1.6

Rewrite $- 1 e^{16}$ as $- e^{16}$ .

$(4 - 2 e^{8} + 2 e^{8} - e^{16}) \frac{1}{1 - 5 e^{16}}$

Step 13.8.2

Add $- 2 e^{8}$ and $2 e^{8}$ .

$(4 + 0 - e^{16}) \frac{1}{1 - 5 e^{16}}$

Step 13.8.3

Add $4$ and $0$ .

$(4 - e^{16}) \frac{1}{1 - 5 e^{16}}$

Step 13.9

Apply the distributive property.

$4 \frac{1}{1 - 5 e^{16}} - e^{16} \frac{1}{1 - 5 e^{16}}$

Step 13.10

Combine $4$ and $\frac{1}{1 - 5 e^{16}}$ .

$\frac{4}{1 - 5 e^{16}} - e^{16} \frac{1}{1 - 5 e^{16}}$

Step 13.11

Combine $\frac{1}{1 - 5 e^{16}}$ and $e^{16}$ .

$\frac{4}{1 - 5 e^{16}} - \frac{e^{16}}{1 - 5 e^{16}}$

Step 13.12

Combine the numerators over the common denominator.

$\frac{4 - e^{16}}{1 - 5 e^{16}}$

Step 13.13

Simplify the numerator.

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Step 13.13.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{2} - e^{16}}{1 - 5 e^{16}}$

Step 13.13.2

Rewrite $e^{16}$ as ${(e^{8})}^{2}$ .

$\frac{2^{2} - {(e^{8})}^{2}}{1 - 5 e^{16}}$

Step 13.13.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = e^{8}$ .

$\frac{(2 + e^{8}) (2 - e^{8})}{1 - 5 e^{16}}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{(2 + e^{8}) (2 - e^{8})}{1 - 5 e^{16}}$

Decimal Form:

$0.19999991 \dots$