Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Move the limit into the exponent.

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

Step 11

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

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

Step 12

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

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

Step 13

Move the limit into the exponent.

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

Step 14

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

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

Step 15

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

Tap for more steps...

Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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

Step 15.4

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

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

Step 16

Simplify the answer.

Tap for more steps...

Step 16.1

Simplify the numerator.

Tap for more steps...

Step 16.1.1

Multiply $- 3$ by $8$ .

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

Step 16.1.2

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

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

Step 16.1.3

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

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

Step 16.1.4

Multiply $3$ by $8$ .

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

Step 16.1.5

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

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

Step 16.1.6

Combine the numerators over the common denominator.

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

Step 16.1.7

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

Tap for more steps...

Step 16.1.7.1

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

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

Step 16.1.7.2

Add $24$ and $24$ .

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

Step 16.2

Simplify the denominator.

Tap for more steps...

Step 16.2.1

Multiply $- 3$ by $8$ .

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

Step 16.2.2

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

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

Step 16.2.3

Combine $4$ and $\frac{1}{e^{24}}$ .

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

Step 16.2.4

Multiply $3$ by $8$ .

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

Step 16.2.5

To write $5 e^{24}$ as a fraction with a common denominator, multiply by $\frac{e^{24}}{e^{24}}$ .

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

Step 16.2.6

Combine the numerators over the common denominator.

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

Step 16.2.7

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

Tap for more steps...

Step 16.2.7.1

Move $e^{24}$ .

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

Step 16.2.7.2

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

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

Step 16.2.7.3

Add $24$ and $24$ .

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

Step 16.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 16.4

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

Tap for more steps...

Step 16.4.1

Cancel the common factor.

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

Step 16.4.2

Rewrite the expression.

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

Step 16.5

Apply the distributive property.

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

Step 16.6

Combine $5$ and $\frac{1}{4 + 5 e^{48}}$ .

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

Step 16.7

Combine $e^{48}$ and $\frac{1}{4 + 5 e^{48}}$ .

$\frac{5}{4 + 5 e^{48}} + \frac{e^{48}}{4 + 5 e^{48}}$

Step 16.8

Combine the numerators over the common denominator.

$\frac{5 + e^{48}}{4 + 5 e^{48}}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{5 + e^{48}}{4 + 5 e^{48}}$

Decimal Form:

$0.2$