Calculus Examples

$lim_{n \to 8} \frac{3^{n + 8}}{\frac{{(n + 8)}^{n + 1}}{\frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{n \to 8} 3^{n + 8}}{lim_{n \to 8} \frac{{(n + 8)}^{n + 1}}{\frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 1.2

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 8}}{lim_{n \to 8} \frac{{(n + 8)}^{n + 1}}{\frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 1.3

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

$\frac{3^{lim_{n \to 8} n + lim_{n \to 8} 8}}{lim_{n \to 8} \frac{{(n + 8)}^{n + 1}}{\frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 1.4

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

$\frac{3^{lim_{n \to 8} n + 8}}{lim_{n \to 8} \frac{{(n + 8)}^{n + 1}}{\frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 1.5

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{lim_{n \to 8} {(n + 8)}^{n + 1}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(n + 8)}^{n + 1}$ as $e^{\ln ({(n + 8)}^{n + 1})}$ .

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{lim_{n \to 8} e^{\ln ({(n + 8)}^{n + 1})}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 2.2

Expand $\ln ({(n + 8)}^{n + 1})$ by moving $n + 1$ outside the logarithm.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{lim_{n \to 8} e^{(n + 1) \ln (n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{lim_{n \to 8} (n + 1) \ln (n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.2

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{lim_{n \to 8} n + 1 \cdot lim_{n \to 8} \ln (n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.3

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.4

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot lim_{n \to 8} \ln (n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.5

Move the limit inside the logarithm.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.6

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + lim_{n \to 8} 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.7

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{lim_{n \to 8} \frac{3^{n + 1}}{{(n + 7)}^{n}}}}$

Step 3.8

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{lim_{n \to 8} 3^{n + 1}}{lim_{n \to 8} {(n + 7)}^{n}}}}$

Step 3.9

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} {(n + 7)}^{n}}}}$

Step 3.10

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + lim_{n \to 8} 1}}{lim_{n \to 8} {(n + 7)}^{n}}}}$

Step 3.11

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} {(n + 7)}^{n}}}}$

Step 4

Use the properties of logarithms to simplify the limit.

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Step 4.1

Rewrite ${(n + 7)}^{n}$ as $e^{\ln ({(n + 7)}^{n})}$ .

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} e^{\ln ({(n + 7)}^{n})}}}}$

Step 4.2

Expand $\ln ({(n + 7)}^{n})$ by moving $n$ outside the logarithm.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{lim_{n \to 8} e^{n \ln (n + 7)}}}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the limit into the exponent.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{e^{lim_{n \to 8} n \ln (n + 7)}}}}$

Step 5.2

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{e^{lim_{n \to 8} n \cdot lim_{n \to 8} \ln (n + 7)}}}}$

Step 5.3

Move the limit inside the logarithm.

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n + 7)}}}}$

Step 5.4

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

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n + lim_{n \to 8} 7)}}}}$

Step 5.5

Evaluate the limit of $7$ which is constant as $n$ approaches $8$ .

$\frac{3^{lim_{n \to 8} n + 8}}{\frac{e^{(lim_{n \to 8} n + 1) \cdot \ln (lim_{n \to 8} n + 8)}}{\frac{3^{lim_{n \to 8} n + 1}}{e^{lim_{n \to 8} n \cdot \ln (lim_{n \to 8} n + 7)}}}}$

Step 6

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

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Step 6.1