Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(\frac{n^{2} + 2}{n^{2} - 3})}^{2 n^{2} + 1}$ as $e^{\ln ({(\frac{n^{2} + 2}{n^{2} - 3})}^{2 n^{2} + 1})}$ .

$lim_{n \to 8} e^{\ln ({(\frac{n^{2} + 2}{n^{2} - 3})}^{2 n^{2} + 1})}$

Step 1.2

Expand $\ln ({(\frac{n^{2} + 2}{n^{2} - 3})}^{2 n^{2} + 1})$ by moving $2 n^{2} + 1$ outside the logarithm.

$lim_{n \to 8} e^{(2 n^{2} + 1) \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{n \to 8} (2 n^{2} + 1) \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.2

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

$e^{lim_{n \to 8} 2 n^{2} + 1 \cdot lim_{n \to 8} \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.3

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

$e^{(lim_{n \to 8} 2 n^{2} + lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.4

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

$e^{(2 lim_{n \to 8} n^{2} + lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.5

Move the exponent $2$ from $n^{2}$ outside the limit using the Limits Power Rule.

$e^{(2 {(lim_{n \to 8} n)}^{2} + lim_{n \to 8} 1) \cdot lim_{n \to 8} \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.6

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

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot lim_{n \to 8} \ln (\frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.7

Move the limit inside the logarithm.

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (lim_{n \to 8} \frac{n^{2} + 2}{n^{2} - 3})}$

Step 2.8

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

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{lim_{n \to 8} n^{2} + 2}{lim_{n \to 8} n^{2} - 3})}$

Step 2.9

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

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{lim_{n \to 8} n^{2} + lim_{n \to 8} 2}{lim_{n \to 8} n^{2} - 3})}$

Step 2.10

Move the exponent $2$ from $n^{2}$ outside the limit using the Limits Power Rule.

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + lim_{n \to 8} 2}{lim_{n \to 8} n^{2} - 3})}$

Step 2.11

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

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + 2}{lim_{n \to 8} n^{2} - 3})}$

Step 2.12

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

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + 2}{lim_{n \to 8} n^{2} - lim_{n \to 8} 3})}$

Step 2.13

Move the exponent $2$ from $n^{2}$ outside the limit using the Limits Power Rule.

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + 2}{{(lim_{n \to 8} n)}^{2} - lim_{n \to 8} 3})}$

Step 2.14

Evaluate the limit of $3$ which is constant as $n$ approaches $8$ .

$e^{(2 {(lim_{n \to 8} n)}^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + 2}{{(lim_{n \to 8} n)}^{2} - 1 \cdot 3})}$

Step 3

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

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Step 3.1

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

$e^{(2 \cdot 8^{2} + 1) \cdot \ln (\frac{{(lim_{n \to 8} n)}^{2} + 2}{{(lim_{n \to 8} n)}^{2} - 1 \cdot 3})}$

Step 3.2

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

$e^{(2 \cdot 8^{2} + 1) \cdot \ln (\frac{8^{2} + 2}{{(lim_{n \to 8} n)}^{2} - 1 \cdot 3})}$

Step 3.3

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

$e^{(2 \cdot 8^{2} + 1) \cdot \ln (\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$e^{(2 \cdot 64 + 1) \cdot \ln (\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}$

Step 4.1.2

Multiply $2$ by $64$ .

$e^{(128 + 1) \cdot \ln (\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}$

Step 4.2

Add $128$ and $1$ .

$e^{129 \cdot \ln (\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}$

Step 4.3

Simplify $129 \cdot \ln (\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})$ by moving $129$ inside the logarithm.

$e^{\ln ({(\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}^{129})}$

Step 4.4

Exponentiation and log are inverse functions.

${(\frac{8^{2} + 2}{8^{2} - 1 \cdot 3})}^{129}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

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

${(\frac{64 + 2}{8^{2} - 1 \cdot 3})}^{129}$

Step 4.5.2

Add $64$ and $2$ .

${(\frac{66}{8^{2} - 1 \cdot 3})}^{129}$

Step 4.6

Simplify the denominator.

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Step 4.6.1

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

${(\frac{66}{64 - 1 \cdot 3})}^{129}$

Step 4.6.2

Multiply $- 1$ by $3$ .

${(\frac{66}{64 - 3})}^{129}$

Step 4.6.3

Subtract $3$ from $64$ .

${(\frac{66}{61})}^{129}$

Step 4.7

Apply the product rule to $\frac{66}{61}$ .

$\frac{66^{129}}{61^{129}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{66^{129}}{61^{129}}$

Decimal Form:

$25919.04337747 \dots$