Calculus Examples

$\frac{lim_{x \to 8} 3 e^{3 x} {(e^{3 x} + 1)}^{\frac{1}{x} - 1}}{x}$

Step 1

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

$\frac{3 lim_{x \to 8} e^{3 x} {(e^{3 x} + 1)}^{\frac{1}{x} - 1}}{x}$

Step 2

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

$\frac{3 lim_{x \to 8} e^{3 x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1}{x} - 1}}{x}$

Step 3

Move the limit into the exponent.

$\frac{3 e^{lim_{x \to 8} 3 x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1}{x} - 1}}{x}$

Step 4

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1}{x} - 1}}{x}$

Step 5

Combine terms.

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Step 5.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1}{x} - 1 \cdot \frac{x}{x}}}{x}$

Step 5.2

Combine $- 1$ and $\frac{x}{x}$ .

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1}{x} + \frac{- x}{x}}}{x}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} {(e^{3 x} + 1)}^{\frac{1 - x}{x}}}{x}$

Step 6

Use the properties of logarithms to simplify the limit.

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Step 6.1

Rewrite ${(e^{3 x} + 1)}^{\frac{1 - x}{x}}$ as $e^{\ln ({(e^{3 x} + 1)}^{\frac{1 - x}{x}})}$ .

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} e^{\ln ({(e^{3 x} + 1)}^{\frac{1 - x}{x}})}}{x}$

Step 6.2

Expand $\ln ({(e^{3 x} + 1)}^{\frac{1 - x}{x}})$ by moving $\frac{1 - x}{x}$ outside the logarithm.

$\frac{3 e^{3 lim_{x \to 8} x} \cdot lim_{x \to 8} e^{\frac{1 - x}{x} \ln (e^{3 x} + 1)}}{x}$

Step 7

Evaluate the limit.

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Step 7.1

Move the limit into the exponent.

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{lim_{x \to 8} \frac{1 - x}{x} \ln (e^{3 x} + 1)}}{x}$

Step 7.2

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{lim_{x \to 8} \frac{1 - x}{x} \cdot lim_{x \to 8} \ln (e^{3 x} + 1)}}{x}$

Step 7.3

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{lim_{x \to 8} 1 - x}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (e^{3 x} + 1)}}{x}$

Step 7.4

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{lim_{x \to 8} 1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (e^{3 x} + 1)}}{x}$

Step 7.5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (e^{3 x} + 1)}}{x}$

Step 7.6

Move the limit inside the logarithm.

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (lim_{x \to 8} e^{3 x} + 1)}}{x}$

Step 7.7

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (lim_{x \to 8} e^{3 x} + lim_{x \to 8} 1)}}{x}$

Step 7.8

Move the limit into the exponent.

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (e^{lim_{x \to 8} 3 x} + lim_{x \to 8} 1)}}{x}$

Step 7.9

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

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (e^{3 lim_{x \to 8} x} + lim_{x \to 8} 1)}}{x}$

Step 7.10

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{3 e^{3 lim_{x \to 8} x} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (e^{3 lim_{x \to 8} x} + 1)}}{x}$

Step 8

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

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Step 8.1

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

$\frac{3 e^{3 \cdot 8} \cdot e^{\frac{1 - lim_{x \to 8} x}{lim_{x \to 8} x} \cdot \ln (e^{3 lim_{x \to 8} x} + 1)}}{x}$

Step 8.2

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

$\frac{3 e^{3 \cdot 8} \cdot e^{\frac{1 - 8}{lim_{x \to 8} x} \cdot \ln (e^{3 lim_{x \to 8} x} + 1)}}{x}$

Step 8.3

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

$\frac{3 e^{3 \cdot 8} \cdot e^{\frac{1 - 8}{8} \cdot \ln (e^{3 lim_{x \to 8} x} + 1)}}{x}$

Step 8.4

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

$\frac{3 e^{3 \cdot 8} \cdot e^{\frac{1 - 8}{8} \cdot \ln (e^{3 \cdot 8} + 1)}}{x}$

Step 9

Multiply $e^{3 \cdot 8}$ by $e^{\frac{1 - 8}{8} \cdot \ln (e^{3 \cdot 8} + 1)}$ by adding the exponents.

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Step 9.1

Move $e^{\frac{1 - 8}{8} \cdot \ln (e^{3 \cdot 8} + 1)}$ .

$\frac{3 (e^{\frac{1 - 8}{8} \cdot \ln (e^{3 \cdot 8} + 1)} e^{3 \cdot 8})}{x}$

Step 9.2

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

$\frac{3 e^{\frac{1 - 8}{8} \cdot \ln (e^{3 \cdot 8} + 1) + 3 \cdot 8}}{x}$

Step 9.3

Subtract $8$ from $1$ .

$\frac{3 e^{\frac{- 7}{8} \cdot \ln (e^{3 \cdot 8} + 1) + 3 \cdot 8}}{x}$

Step 9.4

Multiply $3$ by $8$ .

$\frac{3 e^{\frac{- 7}{8} \cdot \ln (e^{24} + 1) + 3 \cdot 8}}{x}$

Step 9.5

Simplify each term.

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Step 9.5.1

Move the negative in front of the fraction.

$\frac{3 e^{- \frac{7}{8} \cdot \ln (e^{24} + 1) + 3 \cdot 8}}{x}$

Step 9.5.2

Multiply $- \frac{7}{8} \ln (e^{24} + 1)$ .

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Step 9.5.2.1

Reorder $\ln (e^{24} + 1)$ and $\frac{7}{8}$ .

$\frac{3 e^{- (\frac{7}{8} \ln (e^{24} + 1)) + 3 \cdot 8}}{x}$

Step 9.5.2.2

Simplify $\frac{7}{8} \ln (e^{24} + 1)$ by moving $\frac{7}{8}$ inside the logarithm.

$\frac{3 e^{- \ln ({(e^{24} + 1)}^{\frac{7}{8}}) + 3 \cdot 8}}{x}$

Step 9.5.3

Multiply $3$ by $8$ .

$\frac{3 e^{- \ln ({(e^{24} + 1)}^{\frac{7}{8}}) + 24}}{x}$