求該極限？

01-15

$eqalign{ mathop {lim }limits_{x o 0} {(x + {{ ext{e}}^x})^{frac{2}{x}}} cr y = {(x + {{ ext{e}}^x})^{frac{2}{x}}},ln y = frac{2}{x}ln (x + {{ ext{e}}^x}), cr y = {{ ext{e}}^{frac{2}{x}ln (x + {{ ext{e}}^x})}}, cr mathop {lim }limits_{x o 0} {(x + {{ ext{e}}^x})^{frac{2}{x}}} cr = mathop {lim }limits_{x o 0} {{ ext{e}}^{frac{2}{x}ln (x + {{ ext{e}}^x})}} cr = {{ ext{e}}^{2mathop {lim }limits_{x o 0} frac{{ln (x + {{ ext{e}}^x})}}{x}}} cr = {{ ext{e}}^{2mathop {lim }limits_{x o 0} frac{{1 + {{ ext{e}}^x}}}{{x + {{ ext{e}}^x}}}}} cr = {{ ext{e}}^4} cr}$

$lim_{x->0} (x+e^x)^frac{2}{x} = e^2lim_{x->0} (1+frac{x}{e^x})^frac{2}{x} = e^2 lim_{x->0} (1+frac{x}{e^x})^frac{2e^x}{xe^x} = e^2 e^2 = e^4$

獻醜

用第二個重要極限公式結合洛必達法則就可以解。