Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

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

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

Step 7

Move the limit into the exponent.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Move the limit into the exponent.

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

Step 12

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

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

Step 13

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

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

Step 14

Move the limit into the exponent.

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

Step 15

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

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

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

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

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

Step 17

Simplify the answer.

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Step 17.1

Simplify the numerator.

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Step 17.1.1

Multiply $4$ by $8$ .

$\frac{5 e^{32} - 4 e^{- 4 \cdot 8}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.2

Multiply $- 4$ by $8$ .

$\frac{5 e^{32} - 4 e^{- 32}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.3

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

$\frac{5 e^{32} - 4 \frac{1}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.4

Combine $- 4$ and $\frac{1}{e^{32}}$ .

$\frac{5 e^{32} + \frac{- 4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.5

Move the negative in front of the fraction.

$\frac{5 e^{32} - \frac{4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.6

To write $5 e^{32}$ as a fraction with a common denominator, multiply by $\frac{e^{32}}{e^{32}}$ .

$\frac{\frac{5 e^{32} e^{32}}{e^{32}} - \frac{4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.7

Combine the numerators over the common denominator.

$\frac{\frac{5 e^{32} e^{32} - 4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.8

Multiply $e^{32}$ by $e^{32}$ by adding the exponents.

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Step 17.1.8.1

Move $e^{32}$ .

$\frac{\frac{5 (e^{32} e^{32}) - 4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.8.2

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

$\frac{\frac{5 e^{32 + 32} - 4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.1.8.3

Add $32$ and $32$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{3 e^{4 \cdot 8} + 8 e^{- 4 \cdot 8}}$

Step 17.2

Simplify the denominator.

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Step 17.2.1

Multiply $4$ by $8$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{3 e^{32} + 8 e^{- 4 \cdot 8}}$

Step 17.2.2

Multiply $- 4$ by $8$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{3 e^{32} + 8 e^{- 32}}$

Step 17.2.3

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

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{3 e^{32} + 8 \frac{1}{e^{32}}}$

Step 17.2.4

Combine $8$ and $\frac{1}{e^{32}}$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{3 e^{32} + \frac{8}{e^{32}}}$

Step 17.2.5

To write $3 e^{32}$ as a fraction with a common denominator, multiply by $\frac{e^{32}}{e^{32}}$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{\frac{3 e^{32} e^{32}}{e^{32}} + \frac{8}{e^{32}}}$

Step 17.2.6

Combine the numerators over the common denominator.

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{\frac{3 e^{32} e^{32} + 8}{e^{32}}}$

Step 17.2.7

Multiply $e^{32}$ by $e^{32}$ by adding the exponents.

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Step 17.2.7.1

Move $e^{32}$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{\frac{3 (e^{32} e^{32}) + 8}{e^{32}}}$

Step 17.2.7.2

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

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{\frac{3 e^{32 + 32} + 8}{e^{32}}}$

Step 17.2.7.3

Add $32$ and $32$ .

$\frac{\frac{5 e^{64} - 4}{e^{32}}}{\frac{3 e^{64} + 8}{e^{32}}}$

Step 17.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 e^{64} - 4}{e^{32}} \cdot \frac{e^{32}}{3 e^{64} + 8}$

Step 17.4

Cancel the common factor of $e^{32}$ .

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Step 17.4.1

Cancel the common factor.

$\frac{5 e^{64} - 4}{e^{32}} \cdot \frac{e^{32}}{3 e^{64} + 8}$

Step 17.4.2

Rewrite the expression.

$(5 e^{64} - 4) \frac{1}{3 e^{64} + 8}$

Step 17.5

Apply the distributive property.

$5 e^{64} \frac{1}{3 e^{64} + 8} - 4 \frac{1}{3 e^{64} + 8}$

Step 17.6

Multiply $5 e^{64} \frac{1}{3 e^{64} + 8}$ .

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Step 17.6.1

Combine $\frac{1}{3 e^{64} + 8}$ and $5$ .

$\frac{5}{3 e^{64} + 8} e^{64} - 4 \frac{1}{3 e^{64} + 8}$

Step 17.6.2

Combine $\frac{5}{3 e^{64} + 8}$ and $e^{64}$ .

$\frac{5 e^{64}}{3 e^{64} + 8} - 4 \frac{1}{3 e^{64} + 8}$

Step 17.7

Combine $- 4$ and $\frac{1}{3 e^{64} + 8}$ .

$\frac{5 e^{64}}{3 e^{64} + 8} + \frac{- 4}{3 e^{64} + 8}$

Step 17.8

Combine the numerators over the common denominator.

$\frac{5 e^{64} - 4}{3 e^{64} + 8}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{5 e^{64} - 4}{3 e^{64} + 8}$

Decimal Form:

$1.\overline{6}$