Calculus Examples

$lim_{n \to 1000000} {(n^{2} + n)}^{\frac{1}{2 n + 1}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

$lim_{n \to 1000000} e^{\ln ({(n^{2} + n)}^{\frac{1}{2 n + 1}})}$

Step 1.2

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

$lim_{n \to 1000000} e^{\frac{1}{2 n + 1} \ln (n^{2} + n)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{n \to 1000000} \frac{1}{2 n + 1} \ln (n^{2} + n)}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $1000000$ .

$e^{lim_{n \to 1000000} \frac{1}{2 n + 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $1000000$ .

$e^{\frac{lim_{n \to 1000000} 1}{lim_{n \to 1000000} 2 n + 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.4

Evaluate the limit of $1$ which is constant as $n$ approaches $1000000$ .

$e^{\frac{1}{lim_{n \to 1000000} 2 n + 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $1000000$ .

$e^{\frac{1}{lim_{n \to 1000000} 2 n + lim_{n \to 1000000} 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.6

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

$e^{\frac{1}{2 lim_{n \to 1000000} n + lim_{n \to 1000000} 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.7

Evaluate the limit of $1$ which is constant as $n$ approaches $1000000$ .

$e^{\frac{1}{2 lim_{n \to 1000000} n + 1} \cdot lim_{n \to 1000000} \ln (n^{2} + n)}$

Step 2.8

Move the limit inside the logarithm.

$e^{\frac{1}{2 lim_{n \to 1000000} n + 1} \cdot \ln (lim_{n \to 1000000} n^{2} + n)}$

Step 2.9

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $1000000$ .

$e^{\frac{1}{2 lim_{n \to 1000000} n + 1} \cdot \ln (lim_{n \to 1000000} n^{2} + lim_{n \to 1000000} n)}$

Step 2.10

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

$e^{\frac{1}{2 lim_{n \to 1000000} n + 1} \cdot \ln ({(lim_{n \to 1000000} n)}^{2} + lim_{n \to 1000000} n)}$

Step 3

Evaluate the limits by plugging in $1000000$ for all occurrences of $n$ .

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Step 3.1

Evaluate the limit of $n$ by plugging in $1000000$ for $n$ .

$e^{\frac{1}{2 \cdot 1000000 + 1} \cdot \ln ({(lim_{n \to 1000000} n)}^{2} + lim_{n \to 1000000} n)}$

Step 3.2

Evaluate the limit of $n$ by plugging in $1000000$ for $n$ .

$e^{\frac{1}{2 \cdot 1000000 + 1} \cdot \ln (1000000^{2} + lim_{n \to 1000000} n)}$

Step 3.3

Evaluate the limit of $n$ by plugging in $1000000$ for $n$ .

$e^{\frac{1}{2 \cdot 1000000 + 1} \cdot \ln (1000000^{2} + 1000000)}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $2$ by $1000000$ .

$e^{\frac{1}{2000000 + 1} \cdot \ln (1000000^{2} + 1000000)}$

Step 4.1.2

Add $2000000$ and $1$ .

$e^{\frac{1}{2000001} \cdot \ln (1000000^{2} + 1000000)}$

Step 4.2

Raise $1000000$ to the power of $2$ .

$e^{\frac{1}{2000001} \cdot \ln (1000000000000 + 1000000)}$

Step 4.3

Add $1000000000000$ and $1000000$ .

$e^{\frac{1}{2000001} \cdot \ln (1000001000000)}$

Step 4.4

Combine $\frac{1}{2000001}$ and $\ln (1000001000000)$ .

$e^{\frac{\ln (1000001000000)}{2000001}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (1000001000000)}{2000001}}$

Decimal Form:

$1.00001381 \dots$