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Introduction to limits

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By Expert Faculty of Sri Chaitanya

Question

The value of 78.6 $\sqrt{e} \lim_{x \to 0} {\{\frac{{(1 + x)}^{\frac{1}{x}}}{e}\}}^{\frac{1}{x}} i s$

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Solution

$Let P = lim_{x \to 0} {\{\frac{(1 + x)^{1 / x}}{e}\}}^{1 / x}$
$\begin{array}{l} = e^{\lim_{x \to 0} (\frac{{(1 + x)}^{\frac{1}{x}}}{e} - 1) \frac{1}{x}} \\ = e^{\frac{1}{e} lim_{x \to 0} \frac{(1 + x)^{1 / x} - e}{x}} \\ = e^{\frac{1}{e} (- e / 2)} = e^{- 1 / 2} = \frac{1}{\sqrt{e}} (\sin c e lim_{x \to 0} \frac{(1 + x)^{1 / x} - e}{x} = lim_{x \to 0} \frac{e (1 - \frac{x}{2} + \frac{11}{24} x^{2} . .) - e}{x} = \frac{- e}{2}) \end{array}$
$∴ 78 .6 \sqrt{e} lim_{x \to 0} {\{\frac{(1 + x)^{1 / x}}{e}\}}^{1 / x} = 78 .6 \sqrt{e} \cdot \frac{1}{\sqrt{e}} = 78 .6$

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Similar Questions

$If \lim_{x \to 0} {[1 + x + \frac{f (x)}{x}]}^{1 / x} = e^{3}, then the value of \ln (\lim_{x \to 0} {[1 + \frac{f (x)}{x}]}^{1 / x}) is$ ______

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