Calculus Examples

$lim_{n \to 8} \frac{4^{n}}{5^{n + 1}}$

Step 1

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

$\frac{lim_{n \to 8} 4^{n}}{lim_{n \to 8} 5^{n + 1}}$

Step 2

Move the limit into the exponent.

$\frac{4^{lim_{n \to 8} n}}{lim_{n \to 8} 5^{n + 1}}$

Step 3

Move the limit into the exponent.

$\frac{4^{lim_{n \to 8} n}}{5^{lim_{n \to 8} n + 1}}$

Step 4

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

$\frac{4^{lim_{n \to 8} n}}{5^{lim_{n \to 8} n + lim_{n \to 8} 1}}$

Step 5

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

$\frac{4^{lim_{n \to 8} n}}{5^{lim_{n \to 8} n + 1}}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{4^{8}}{5^{lim_{n \to 8} n + 1}}$

Step 6.2

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

$\frac{4^{8}}{5^{8 + 1}}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Raise $4$ to the power of $8$ .

$\frac{65536}{5^{8 + 1}}$

Step 7.2

Simplify the denominator.

Tap for more steps...

Step 7.2.1

Add $8$ and $1$ .

$\frac{65536}{5^{9}}$

Step 7.2.2

Raise $5$ to the power of $9$ .

$\frac{65536}{1953125}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{65536}{1953125}$

Decimal Form:

$0.03355443 \dots$