The value of lim n -n -m nn1 - rA- emB- e - m - emD- noneof these

The correct option is C 1/emL=lim_n →∞1/m(1/n.2/n.3/n.....n/n ) ^1/n In L=lim_n →∞ [ ln (1/m )+1/n(ln1/n+ln2/n+...+ln n/n ) ] ln m+lim_n →∞1/n∑_r=1^nln (r/n ...

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Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

The value of ...

Question

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

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\frac{e}{m}

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\frac{1}{e m}

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n o n e o f t h e s e

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Solution

The correct option is C $\frac{1}{e m}$
$L = {lim}_{n \to \infty} \frac{1}{m} {(\frac{1}{n} . \frac{2}{n} . \frac{3}{n} . . . . . \frac{n}{n})}^{1 / n}$
In $L = {lim}_{n \to \infty} [l n (\frac{1}{m}) + \frac{1}{n} (l n \frac{1}{n} + l n \frac{2}{n} + . . . + l n \frac{n}{n})]$
$- l n m + {lim}_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} l n (\frac{r}{n}) = - l n m + \int_{0}^{1} l n x d x = - l n m - 1 = l n (\frac{1}{e m})$
$∴ L = \frac{1}{e m}$

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The value of limn→∞(n!(mn)n)1/n is

The correct option is C 1emL=limn→∞1m(1n.2n.3n.....nn)1/n In L=limn→∞[ln(1m)+1n(ln1n+ln2n+...+lnnn)] −ln m+limn→∞1n∑nr=1ln(rn)=−ln m+∫10ln xdx=−ln m−1=ln(1em) ∴L=1em

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