Calculus Examples

$lim_{n \to 8} \frac{2^{n + 1} + 3^{n + 1}}{2^{n} + 3^{n}}$

Step 1

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

$\frac{lim_{n \to 8} 2^{n + 1} + 3^{n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 2

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

$\frac{lim_{n \to 8} 2^{n + 1} + lim_{n \to 8} 3^{n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 3

Move the limit into the exponent.

$\frac{2^{lim_{n \to 8} n + 1} + lim_{n \to 8} 3^{n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 4

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

$\frac{2^{lim_{n \to 8} n + lim_{n \to 8} 1} + lim_{n \to 8} 3^{n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 5

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

$\frac{2^{lim_{n \to 8} n + 1} + lim_{n \to 8} 3^{n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 6

Move the limit into the exponent.

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 7

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

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + lim_{n \to 8} 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 8

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

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} 2^{n} + 3^{n}}$

Step 9

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

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} 2^{n} + lim_{n \to 8} 3^{n}}$

Step 10

Move the limit into the exponent.

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + 1}}{2^{lim_{n \to 8} n} + lim_{n \to 8} 3^{n}}$

Step 11

Move the limit into the exponent.

$\frac{2^{lim_{n \to 8} n + 1} + 3^{lim_{n \to 8} n + 1}}{2^{lim_{n \to 8} n} + 3^{lim_{n \to 8} n}}$

Step 12

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

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Step 12.1

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

$\frac{2^{8 + 1} + 3^{lim_{n \to 8} n + 1}}{2^{lim_{n \to 8} n} + 3^{lim_{n \to 8} n}}$

Step 12.2

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

$\frac{2^{8 + 1} + 3^{8 + 1}}{2^{lim_{n \to 8} n} + 3^{lim_{n \to 8} n}}$

Step 12.3

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

$\frac{2^{8 + 1} + 3^{8 + 1}}{2^{8} + 3^{lim_{n \to 8} n}}$

Step 12.4

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

$\frac{2^{8 + 1} + 3^{8 + 1}}{2^{8} + 3^{8}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Add $8$ and $1$ .

$\frac{2^{9} + 3^{8 + 1}}{2^{8} + 3^{8}}$

Step 13.1.2

Raise $2$ to the power of $9$ .

$\frac{512 + 3^{8 + 1}}{2^{8} + 3^{8}}$

Step 13.1.3

Add $8$ and $1$ .

$\frac{512 + 3^{9}}{2^{8} + 3^{8}}$

Step 13.1.4

Raise $3$ to the power of $9$ .

$\frac{512 + 19683}{2^{8} + 3^{8}}$

Step 13.1.5

Add $512$ and $19683$ .

$\frac{20195}{2^{8} + 3^{8}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raise $2$ to the power of $8$ .

$\frac{20195}{256 + 3^{8}}$

Step 13.2.2

Raise $3$ to the power of $8$ .

$\frac{20195}{256 + 6561}$

Step 13.2.3

Add $256$ and $6561$ .

$\frac{20195}{6817}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{20195}{6817}$

Decimal Form:

$2.96244682 \dots$