Precalculus Examples

$lim_{n \to 8} \frac{n!}{n^{n}}$

Step 1

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

$\frac{lim_{n \to 8} n!}{lim_{n \to 8} n^{n}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

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

Step 3.2

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

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

Step 3.3

Move the limit inside the logarithm.

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

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 $n$ .

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

Step 4.2

Expand $8!$ to $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3

Multiply $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 4.3.1

Multiply $8$ by $7$ .

$\frac{56 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.2

Multiply $56$ by $6$ .

$\frac{336 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.3

Multiply $336$ by $5$ .

$\frac{1680 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.4

Multiply $1680$ by $4$ .

$\frac{6720 \cdot 3 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.5

Multiply $6720$ by $3$ .

$\frac{20160 \cdot 2 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.6

Multiply $20160$ by $2$ .

$\frac{40320 \cdot 1}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.3.7

Multiply $40320$ by $1$ .

$\frac{40320}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n)}}$

Step 4.4

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

$\frac{40320}{e^{8 \cdot \ln (lim_{n \to 8} n)}}$

Step 4.5

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

$\frac{40320}{e^{8 \cdot \ln (8)}}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Simplify $8 \cdot \ln (8)$ by moving $8$ inside the logarithm.

$\frac{40320}{e^{\ln (8^{8})}}$

Step 5.1.2

Exponentiation and log are inverse functions.

$\frac{40320}{8^{8}}$

Step 5.1.3

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

$\frac{40320}{16777216}$

Step 5.2

Cancel the common factor of $40320$ and $16777216$ .

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Step 5.2.1

Factor $128$ out of $40320$ .

$\frac{128 (315)}{16777216}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $128$ out of $16777216$ .

$\frac{128 \cdot 315}{128 \cdot 131072}$

Step 5.2.2.2

Cancel the common factor.

$\frac{128 \cdot 315}{128 \cdot 131072}$

Step 5.2.2.3

Rewrite the expression.

$\frac{315}{131072}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{315}{131072}$

Decimal Form:

$0.00240325 \dots$