Calculus Examples

$lim_{x \to - 8} e^{\frac{5 e^{3 x} - 4 e^{- 3 x}}{3 e^{3 x} + 8 e^{- 3 x}}}$

Step 1

Move the limit into the exponent.

$e^{lim_{x \to - 8} \frac{5 e^{3 x} - 4 e^{- 3 x}}{3 e^{3 x} + 8 e^{- 3 x}}}$

Step 2

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

$e^{\frac{lim_{x \to - 8} 5 e^{3 x} - 4 e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 3

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

$e^{\frac{lim_{x \to - 8} 5 e^{3 x} - lim_{x \to - 8} 4 e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 4

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

$e^{\frac{5 lim_{x \to - 8} e^{3 x} - lim_{x \to - 8} 4 e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 5

Move the limit into the exponent.

$e^{\frac{5 e^{lim_{x \to - 8} 3 x} - lim_{x \to - 8} 4 e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 6

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - lim_{x \to - 8} 4 e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 7

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 lim_{x \to - 8} e^{- 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 8

Move the limit into the exponent.

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{lim_{x \to - 8} - 3 x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 9

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{lim_{x \to - 8} 3 e^{3 x} + 8 e^{- 3 x}}}$

Step 10

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{lim_{x \to - 8} 3 e^{3 x} + lim_{x \to - 8} 8 e^{- 3 x}}}$

Step 11

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 lim_{x \to - 8} e^{3 x} + lim_{x \to - 8} 8 e^{- 3 x}}}$

Step 12

Move the limit into the exponent.

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{lim_{x \to - 8} 3 x} + lim_{x \to - 8} 8 e^{- 3 x}}}$

Step 13

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{3 lim_{x \to - 8} x} + lim_{x \to - 8} 8 e^{- 3 x}}}$

Step 14

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{3 lim_{x \to - 8} x} + 8 lim_{x \to - 8} e^{- 3 x}}}$

Step 15

Move the limit into the exponent.

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{3 lim_{x \to - 8} x} + 8 e^{lim_{x \to - 8} - 3 x}}}$

Step 16

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

$e^{\frac{5 e^{3 lim_{x \to - 8} x} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{3 lim_{x \to - 8} x} + 8 e^{- 3 lim_{x \to - 8} x}}}$

Step 17

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

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Step 17.1

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

$e^{\frac{5 e^{3 \cdot - 8} - 4 e^{- 3 lim_{x \to - 8} x}}{3 e^{3 lim_{x \to - 8} x} + 8 e^{- 3 lim_{x \to - 8} x}}}$

Step 17.2

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

$e^{\frac{5 e^{3 \cdot - 8} - 4 e^{- 3 \cdot - 8}}{3 e^{3 lim_{x \to - 8} x} + 8 e^{- 3 lim_{x \to - 8} x}}}$

Step 17.3

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

$e^{\frac{5 e^{3 \cdot - 8} - 4 e^{- 3 \cdot - 8}}{3 e^{3 \cdot - 8} + 8 e^{- 3 lim_{x \to - 8} x}}}$

Step 17.4

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

$e^{\frac{5 e^{3 \cdot - 8} - 4 e^{- 3 \cdot - 8}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18

Simplify the answer.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

Multiply $3$ by $- 8$ .

$e^{\frac{5 e^{- 24} - 4 e^{- 3 \cdot - 8}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.2

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

$e^{\frac{5 \frac{1}{e^{24}} - 4 e^{- 3 \cdot - 8}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.3

Combine $5$ and $\frac{1}{e^{24}}$ .

$e^{\frac{\frac{5}{e^{24}} - 4 e^{- 3 \cdot - 8}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.4

Multiply $- 3$ by $- 8$ .

$e^{\frac{\frac{5}{e^{24}} - 4 e^{24}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.5

To write $- 4 e^{24}$ as a fraction with a common denominator, multiply by $\frac{e^{24}}{e^{24}}$ .

$e^{\frac{\frac{5}{e^{24}} - 4 e^{24} \cdot \frac{e^{24}}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.6

Combine $- 4 e^{24}$ and $\frac{e^{24}}{e^{24}}$ .

$e^{\frac{\frac{5}{e^{24}} + \frac{- 4 e^{24} e^{24}}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.7

Combine the numerators over the common denominator.

$e^{\frac{\frac{5 - 4 e^{24} e^{24}}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.8

Multiply $e^{24}$ by $e^{24}$ by adding the exponents.

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Step 18.1.8.1

Move $e^{24}$ .

$e^{\frac{\frac{5 - 4 (e^{24} e^{24})}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.8.2

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

$e^{\frac{\frac{5 - 4 e^{24 + 24}}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.1.8.3

Add $24$ and $24$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{3 e^{3 \cdot - 8} + 8 e^{- 3 \cdot - 8}}}$

Step 18.2

Simplify the denominator.

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Step 18.2.1

Multiply $3$ by $- 8$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{3 e^{- 24} + 8 e^{- 3 \cdot - 8}}}$

Step 18.2.2

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

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{3 \frac{1}{e^{24}} + 8 e^{- 3 \cdot - 8}}}$

Step 18.2.3

Combine $3$ and $\frac{1}{e^{24}}$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3}{e^{24}} + 8 e^{- 3 \cdot - 8}}}$

Step 18.2.4

Multiply $- 3$ by $- 8$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3}{e^{24}} + 8 e^{24}}}$

Step 18.2.5

To write $8 e^{24}$ as a fraction with a common denominator, multiply by $\frac{e^{24}}{e^{24}}$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3}{e^{24}} + \frac{8 e^{24} e^{24}}{e^{24}}}}$

Step 18.2.6

Combine the numerators over the common denominator.

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3 + 8 e^{24} e^{24}}{e^{24}}}}$

Step 18.2.7

Multiply $e^{24}$ by $e^{24}$ by adding the exponents.

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Step 18.2.7.1

Move $e^{24}$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3 + 8 (e^{24} e^{24})}{e^{24}}}}$

Step 18.2.7.2

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

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3 + 8 e^{24 + 24}}{e^{24}}}}$

Step 18.2.7.3

Add $24$ and $24$ .

$e^{\frac{\frac{5 - 4 e^{48}}{e^{24}}}{\frac{3 + 8 e^{48}}{e^{24}}}}$

Step 18.3

Multiply the numerator by the reciprocal of the denominator.

$e^{\frac{5 - 4 e^{48}}{e^{24}} \cdot \frac{e^{24}}{3 + 8 e^{48}}}$

Step 18.4

Cancel the common factor of $e^{24}$ .

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Step 18.4.1

Cancel the common factor.

$e^{\frac{5 - 4 e^{48}}{e^{24}} \cdot \frac{e^{24}}{3 + 8 e^{48}}}$

Step 18.4.2

Rewrite the expression.

$e^{(5 - 4 e^{48}) \frac{1}{3 + 8 e^{48}}}$

Step 18.5

Apply the distributive property.

$e^{5 \frac{1}{3 + 8 e^{48}} - 4 e^{48} \frac{1}{3 + 8 e^{48}}}$

Step 18.6

Combine $5$ and $\frac{1}{3 + 8 e^{48}}$ .

$e^{\frac{5}{3 + 8 e^{48}} - 4 e^{48} \frac{1}{3 + 8 e^{48}}}$

Step 18.7

Multiply $- 4 e^{48} \frac{1}{3 + 8 e^{48}}$ .

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Step 18.7.1

Combine $\frac{1}{3 + 8 e^{48}}$ and $- 4$ .

$e^{\frac{5}{3 + 8 e^{48}} + \frac{- 4}{3 + 8 e^{48}} e^{48}}$

Step 18.7.2

Combine $\frac{- 4}{3 + 8 e^{48}}$ and $e^{48}$ .

$e^{\frac{5}{3 + 8 e^{48}} + \frac{- 4 e^{48}}{3 + 8 e^{48}}}$

Step 18.8

Combine the numerators over the common denominator.

$e^{\frac{5 - 4 e^{48}}{3 + 8 e^{48}}}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{5 - 4 e^{48}}{3 + 8 e^{48}}}$

Decimal Form:

$0.60653065 \dots$