The value of limn-nn-1-n is equal to

Let L=lim_n→∞n(n!)^1/n =lim_n→∞n(1·2·3⋯ n)^1/n =lim_n→∞1(1n·2n·3n⋯nn)^1/n Taking ln on both sides, we get ln L= lim_n→∞1n[ln(1n)+ln(2n)+ln(3n)+…+ln(nn)] ⇒ln L= ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

The value of ...

Question

The value of $lim n \to \infty \frac{n}{(n!)^{1 / n}}$ is equal to

e

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\frac{1}{e}

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1

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1 \int 0 ln x d x

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Solution

The correct option is A $e$
Let $L = lim n \to \infty \frac{n}{(n!)^{1 / n}}$
$= lim n \to \infty \frac{n}{(1 \cdot 2 \cdot 3 \dots n)^{1 / n}}$
$= lim n \to \infty \frac{1}{{(\frac{1}{n} \cdot \frac{2}{n} \cdot \frac{3}{n} \dots \frac{n}{n})}^{1 / n}}$
Taking $ln$ on both sides, we get
$ln L = - lim n \to \infty \frac{1}{n} [ln (\frac{1}{n}) + ln (\frac{2}{n}) + ln (\frac{3}{n}) + \dots + ln (\frac{n}{n})]$
$\Rightarrow ln L = - lim n \to \infty n \sum r = 1 \frac{1}{n} ln (\frac{r}{n})$
$\Rightarrow ln L = - 1 \int 0 (ln x) d x$
$\Rightarrow ln L = 1$
$∴ L = e$

Suggest Corrections

The value of limn→∞n(n!)1/n is equal to

The correct option is A eLet L=limn→∞n(n!)1/n =limn→∞n(1⋅2⋅3⋯n)1/n =limn→∞1(1n⋅2n⋅3n⋯nn)1/n Taking ln on both sides, we get lnL=−limn→∞1n[ln(1n)+ln(2n)+ln(3n)+⋯+ln(nn)] ⇒lnL=−limn→∞n∑r=11nln(rn) ⇒lnL=−1∫0(lnx)dx ⇒lnL=1 ∴L=e

The value of $lim n \to \infty \frac{n}{(n!)^{1 / n}}$ is equal to