The limit of $((1+x)^{1/x} - e+ ex/2)/x^2$ as $x\to 0$ $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$


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*by directly substituting $x=0$ i got $\infty$

*by using L-H's rule, i got $-1/8$


the given options are
$a)\frac{24e}{11}$  
$b)\frac{11e}{24}$  
$c)\frac{e}{11}$  
$d)\frac{e}{24}$  
may be the question would be 
$$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$

Probably you have a typo. Notice that by the Taylor series we have
$$(1+x)^{\color{red}{\frac1x}}=e-\frac{ex}2+\frac{11e x^2}{24}+\mathcal O(x^3)$$
