Calculus Examples

$lim_{n \to \infty} \sqrt[n]{| a_{n} |}$

Step 1

Evaluate the limit.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[n]{| a_{n} |}$ as ${| a_{n} |}^{\frac{1}{n}}$ .

$lim_{n \to \infty} {| a_{n} |}^{\frac{1}{n}}$

Step 1.2

Move the limit into the exponent.

${| a_{n} |}^{lim_{n \to \infty} \frac{1}{n}}$

Step 2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{n}$ approaches $0$ .

${| a_{n} |}^{0}$

Step 3

Simplify the answer.

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Step 3.1

Remove the absolute value in ${| a_{n} |}^{0}$ because exponentiations with even powers are always positive.

${a_{n}}^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

$1$