Question

Evaluate $\underset{x\to 0}{lim}\frac{(1+x{)}^{1/x}-e+\frac{1}{2}ex}{x^2}$

Answer 1

Note: Expansion of $(1+x{)}^{1/x}=e(1-\frac{x}{2}+\frac{11x^2}{24}+o(x^3\left)\right)$
$\therefore \underset{x\to 0}{lim}\frac{(1+x{)}^{1/x}-e+\frac{1}{2}ex}{x^2}=\underset{x\to 0}{lim}\frac{e(1-\frac{x}{2}+\frac{11x^2}{24}+o(x^3\left)\right)-e+\frac{1}{2}ex}{x^2}$
$=\underset{x\to 0}{lim}\frac{e-\frac{ex}{2}+\frac{11e{x}^2}{24}+o\left(x^3\right)-e+\frac{1}{2}ex}{x^2}$
$=\underset{x\to 0}{lim}\frac{\frac{11e{x}^2}{24}+o\left(x^3\right)}{x^2}$
$=\underset{x\to 0}{lim}\frac{11e}{24}+o\left(x\right)$
$=\frac{11e}{24}$
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Where $o\left(x^n\right)$ is used to denote a power series where the least power of $x$ is $n$.