Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

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

Step 3.2

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

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

Step 3.3

Move the limit inside the logarithm.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Exponentiation and log are inverse functions.

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

Step 5.3

Add $8$ and $1$ .

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

Step 5.4

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

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

Step 5.5

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

$\frac{43046721}{8^{8}}$

Step 5.6

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

$\frac{43046721}{16777216}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{43046721}{16777216}$

Decimal Form:

$2.56578451 \dots$