Calculus Examples

$lim_{x \to (- 8)} \frac{4 + 6 e^{2 x}}{5 - 9 e^{3 x}}$

Step 1

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

$\frac{lim_{x \to (- 8)} 4 + 6 e^{2 x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 2

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

$\frac{lim_{x \to (- 8)} 4 + lim_{x \to (- 8)} 6 e^{2 x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $(- 8)$ .

$\frac{4 + lim_{x \to (- 8)} 6 e^{2 x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 + 6 lim_{x \to (- 8)} e^{2 x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 5

Move the limit into the exponent.

$\frac{4 + 6 e^{lim_{x \to (- 8)} 2 x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{lim_{x \to (- 8)} 5 - 9 e^{3 x}}$

Step 7

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

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{lim_{x \to (- 8)} 5 - lim_{x \to (- 8)} 9 e^{3 x}}$

Step 8

Evaluate the limit of $5$ which is constant as $x$ approaches $(- 8)$ .

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{5 - lim_{x \to (- 8)} 9 e^{3 x}}$

Step 9

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{5 - 9 lim_{x \to (- 8)} e^{3 x}}$

Step 10

Move the limit into the exponent.

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{5 - 9 e^{lim_{x \to (- 8)} 3 x}}$

Step 11

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

$\frac{4 + 6 e^{2 lim_{x \to (- 8)} x}}{5 - 9 e^{3 lim_{x \to (- 8)} x}}$

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 $x$ .

$\frac{4 + 6 e^{2 (- 8)}}{5 - 9 e^{3 lim_{x \to (- 8)} x}}$

Step 12.2

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

$\frac{4 + 6 e^{2 (- 8)}}{5 - 9 e^{3 (- 8)}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Multiply $2$ by $- 8$ .

$\frac{4 + 6 e^{- 16}}{5 - 9 e^{3 \cdot - 8}}$

Step 13.1.2

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

$\frac{4 + 6 \frac{1}{e^{16}}}{5 - 9 e^{3 \cdot - 8}}$

Step 13.1.3

Combine $6$ and $\frac{1}{e^{16}}$ .

$\frac{4 + \frac{6}{e^{16}}}{5 - 9 e^{3 \cdot - 8}}$

Step 13.1.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{e^{16}}{e^{16}}$ .

$\frac{\frac{4 e^{16}}{e^{16}} + \frac{6}{e^{16}}}{5 - 9 e^{3 \cdot - 8}}$

Step 13.1.5

Combine the numerators over the common denominator.

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{5 - 9 e^{3 \cdot - 8}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $3$ by $- 8$ .

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{5 - 9 e^{- 24}}$

Step 13.2.2

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

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{5 - 9 \frac{1}{e^{24}}}$

Step 13.2.3

Combine $- 9$ and $\frac{1}{e^{24}}$ .

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{5 + \frac{- 9}{e^{24}}}$

Step 13.2.4

Move the negative in front of the fraction.

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{5 - \frac{9}{e^{24}}}$

Step 13.2.5

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

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{\frac{5 e^{24}}{e^{24}} - \frac{9}{e^{24}}}$

Step 13.2.6

Combine the numerators over the common denominator.

$\frac{\frac{4 e^{16} + 6}{e^{16}}}{\frac{5 e^{24} - 9}{e^{24}}}$

Step 13.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 e^{16} + 6}{e^{16}} \cdot \frac{e^{24}}{5 e^{24} - 9}$

Step 13.4

Cancel the common factor of $e^{16}$ .

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Step 13.4.1

Factor $e^{16}$ out of $e^{24}$ .

$\frac{4 e^{16} + 6}{e^{16}} \cdot \frac{e^{16} e^{8}}{5 e^{24} - 9}$

Step 13.4.2

Cancel the common factor.

$\frac{4 e^{16} + 6}{e^{16}} \cdot \frac{e^{16} e^{8}}{5 e^{24} - 9}$

Step 13.4.3

Rewrite the expression.

$(4 e^{16} + 6) \frac{e^{8}}{5 e^{24} - 9}$

Step 13.5

Apply the distributive property.

$4 e^{16} \frac{e^{8}}{5 e^{24} - 9} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.6

Multiply $4 e^{16} \frac{e^{8}}{5 e^{24} - 9}$ .

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Step 13.6.1

Combine $\frac{e^{8}}{5 e^{24} - 9}$ and $4$ .

$\frac{e^{8} \cdot 4}{5 e^{24} - 9} e^{16} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.6.2

Combine $\frac{e^{8} \cdot 4}{5 e^{24} - 9}$ and $e^{16}$ .

$\frac{e^{8} \cdot 4 e^{16}}{5 e^{24} - 9} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.6.3

Multiply $e^{8}$ by $e^{16}$ by adding the exponents.

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Step 13.6.3.1

Move $e^{16}$ .

$\frac{e^{16} e^{8} \cdot 4}{5 e^{24} - 9} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.6.3.2

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

$\frac{e^{16 + 8} \cdot 4}{5 e^{24} - 9} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.6.3.3

Add $16$ and $8$ .

$\frac{e^{24} \cdot 4}{5 e^{24} - 9} + 6 \frac{e^{8}}{5 e^{24} - 9}$

Step 13.7

Combine $6$ and $\frac{e^{8}}{5 e^{24} - 9}$ .

$\frac{e^{24} \cdot 4}{5 e^{24} - 9} + \frac{6 e^{8}}{5 e^{24} - 9}$

Step 13.8

Combine the numerators over the common denominator.

$\frac{e^{24} \cdot 4 + 6 e^{8}}{5 e^{24} - 9}$

Step 13.9

Move $4$ to the left of $e^{24}$ .

$\frac{4 e^{24} + 6 e^{8}}{5 e^{24} - 9}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{4 e^{24} + 6 e^{8}}{5 e^{24} - 9}$

Decimal Form:

$0.80000013 \dots$