Calculus Examples

$lim_{n \to 8} {(\frac{n - 3}{n})}^{n}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(\frac{n - 3}{n})}^{n}$ as $e^{\ln ({(\frac{n - 3}{n})}^{n})}$ .

$lim_{n \to 8} e^{\ln ({(\frac{n - 3}{n})}^{n})}$

Step 1.2

Expand $\ln ({(\frac{n - 3}{n})}^{n})$ by moving $n$ outside the logarithm.

$lim_{n \to 8} e^{n \ln (\frac{n - 3}{n})}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{n \to 8} n \ln (\frac{n - 3}{n})}$

Step 2.2

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

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

Step 2.3

Move the limit inside the logarithm.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Evaluate the limit of $3$ which is constant as $n$ approaches $8$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify $8 \cdot \ln (\frac{8 - 1 \cdot 3}{8})$ by moving $8$ inside the logarithm.

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

Step 4.2

Exponentiation and log are inverse functions.

${(\frac{8 - 1 \cdot 3}{8})}^{8}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Multiply $- 1$ by $3$ .

${(\frac{8 - 3}{8})}^{8}$

Step 4.3.2

Subtract $3$ from $8$ .

${(\frac{5}{8})}^{8}$

Step 4.4

Apply the product rule to $\frac{5}{8}$ .

$\frac{5^{8}}{8^{8}}$

Step 4.5

Raise $5$ to the power of $8$ .

$\frac{390625}{8^{8}}$

Step 4.6

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

$\frac{390625}{16777216}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{390625}{16777216}$

Decimal Form:

$0.02328306 \dots$