Calculus Examples

$lim_{x \to 8} \frac{48^{x}}{x!}$

Step 1

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

$\frac{lim_{x \to 8} 48^{x}}{lim_{x \to 8} x!}$

Step 2

Move the limit into the exponent.

$\frac{48^{lim_{x \to 8} x}}{lim_{x \to 8} x!}$

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 $x$ .

$\frac{48^{8}}{lim_{x \to 8} x!}$

Step 3.2

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

$\frac{48^{8}}{8!}$

Step 3.3

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

$\frac{48^{8}}{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4

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

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Step 3.4.1

Multiply $8$ by $7$ .

$\frac{48^{8}}{56 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply $56$ by $6$ .

$\frac{48^{8}}{336 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.3

Multiply $336$ by $5$ .

$\frac{48^{8}}{1680 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.4

Multiply $1680$ by $4$ .

$\frac{48^{8}}{6720 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.5

Multiply $6720$ by $3$ .

$\frac{48^{8}}{20160 \cdot 2 \cdot 1}$

Step 3.4.6

Multiply $20160$ by $2$ .

$\frac{48^{8}}{40320 \cdot 1}$

Step 3.4.7

Multiply $40320$ by $1$ .

$\frac{48^{8}}{40320}$

Step 4

Simplify the answer.

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Step 4.1

Raise $48$ to the power of $8$ .

$\frac{28179280429056}{40320}$

Step 4.2

Cancel the common factor of $28179280429056$ and $40320$ .

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Step 4.2.1

Factor $1152$ out of $28179280429056$ .

$\frac{1152 (24461180928)}{40320}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $1152$ out of $40320$ .

$\frac{1152 \cdot 24461180928}{1152 \cdot 35}$

Step 4.2.2.2

Cancel the common factor.

$\frac{1152 \cdot 24461180928}{1152 \cdot 35}$

Step 4.2.2.3

Rewrite the expression.

$\frac{24461180928}{35}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{24461180928}{35}$

Decimal Form:

$698890883.657143$