The correct option is B e Let #160; y #160; = lim _ n→∞ #160; ( e^ n /π #160; #160; ) #160; ^ 1 / n #160; #160;Applying loga ...

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Logarithmic Differentiation

limn→∞en/π1/n...

Question

$lim n \to \infty {(\frac{e^{n}}{π})}^{1 / n} =$

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e^{2}

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e

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Solution

The correct option is B $e$
Let $y$ = $lim n \to \infty {(\frac{e^{n}}{π})}^{\frac{1}{n}}$

Applying logarithm on both sides

$log y$ = $lim n \to \infty log {(\frac{e^{n}}{π})}^{\frac{1}{n}} = lim n \to \infty \frac{1}{n} log (\frac{e^{n}}{π})$

$log y$ = ${lim}_{n \to \infty} \frac{1}{n} (n log e - log π)$

By L'Hospital Rule

$log y = log e$

$\Rightarrow y = e$

$∴ lim n \to \infty {(\frac{e^{n}}{π})}^{\frac{1}{n}} = e$

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