Calculus Examples

$lim_{n \to 8} \frac{3^{2 n}}{4^{n}}$

Step 1

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

$\frac{lim_{n \to 8} 3^{2 n}}{lim_{n \to 8} 4^{n}}$

Step 2

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} 2 n}}{lim_{n \to 8} 4^{n}}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $n$ .

$\frac{3^{2 lim_{n \to 8} n}}{lim_{n \to 8} 4^{n}}$

Step 4

Move the limit into the exponent.

$\frac{3^{2 lim_{n \to 8} n}}{4^{lim_{n \to 8} n}}$

Step 5

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

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Step 5.1

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

$\frac{3^{2 \cdot 8}}{4^{lim_{n \to 8} n}}$

Step 5.2

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

$\frac{3^{2 \cdot 8}}{4^{8}}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $2$ by $8$ .

$\frac{3^{16}}{4^{8}}$

Step 6.1.2

Raise $3$ to the power of $16$ .

$\frac{43046721}{4^{8}}$

Step 6.2

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

$\frac{43046721}{65536}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{43046721}{65536}$

Decimal Form:

$656.84083557 \dots$