Calculus Examples

$lim_{n \to 8} \frac{3^{n + 2}}{5^{n}}$

Step 1

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

$\frac{lim_{n \to 8} 3^{n + 2}}{lim_{n \to 8} 5^{n}}$

Step 2

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 2}}{lim_{n \to 8} 5^{n}}$

Step 3

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

$\frac{3^{lim_{n \to 8} n + lim_{n \to 8} 2}}{lim_{n \to 8} 5^{n}}$

Step 4

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

$\frac{3^{lim_{n \to 8} n + 2}}{lim_{n \to 8} 5^{n}}$

Step 5

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 2}}{5^{lim_{n \to 8} n}}$

Step 6

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

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Step 6.1

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

$\frac{3^{8 + 2}}{5^{lim_{n \to 8} n}}$

Step 6.2

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

$\frac{3^{8 + 2}}{5^{8}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Add $8$ and $2$ .

$\frac{3^{10}}{5^{8}}$

Step 7.1.2

Raise $3$ to the power of $10$ .

$\frac{59049}{5^{8}}$

Step 7.2

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

$\frac{59049}{390625}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{59049}{390625}$

Decimal Form:

$0.15116544$