Calculus Examples

$lim_{n \to 8} n^{\frac{1}{n}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Move the limit inside the logarithm.

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

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 $n$ .

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

Step 3.2

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

$e^{\frac{1}{8} \cdot \ln (8)}$

Step 4

Combine $\frac{1}{8}$ and $\ln (8)$ .

$e^{\frac{\ln (8)}{8}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (8)}{8}}$

Decimal Form:

$1.29683955 \dots$