The value of limx→0 loge⁡(1+x)−xx(1+x)1/x−e equals to

A
$e^{e}$
B
$e$
C
$1 / e$
D
1

Solution:

We have $lim_{x \to 0} \frac{\log_{e} (1 + x) - x}{x \{(1 + x)^{1 / x} - e\}}$

$= lim_{x \to 0} \frac{\log_{e} (1 + x) - x}{x \{e^{1 / x \log (1 + x)} - e\}}$

$= lim_{x \to 0} \frac{- \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots \dots}{x (e^{1 - \frac{x}{2} + \frac{x^{2}}{3} - \frac{x^{3}}{4}} \dots - e)}$

$= \frac{1}{e}$

The value of $lim_{x \to 0} \frac{\log_{e} (1 + x) - x}{x \{(1 + x)^{1 / x} - e\}}$ equals to

Solution:

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