Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

Move the limit inside the logarithm.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify $8 \cdot \ln (\frac{8}{8 + 1})$ by moving $8$ inside the logarithm.

$e^{\ln ({(\frac{8}{8 + 1})}^{8})}$

Step 4.2

Exponentiation and log are inverse functions.

${(\frac{8}{8 + 1})}^{8}$

Step 4.3

Add $8$ and $1$ .

${(\frac{8}{9})}^{8}$

Step 4.4

Apply the product rule to $\frac{8}{9}$ .

$\frac{8^{8}}{9^{8}}$

Step 4.5

Raise $8$ to the power of $8$ .

$\frac{16777216}{9^{8}}$

Step 4.6

Raise $9$ to the power of $8$ .

$\frac{16777216}{43046721}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{16777216}{43046721}$

Decimal Form:

$0.38974434 \dots$