计算 lim(x->0)(1+1/2x)^x

题目

计算 lim(x->0)(1+1/2x)^x

答案

解法一：原式=lim(x->0){(1+1/2x)^[(2/x)(x²/2)]}
=[lim(x->0){(1+1/2x)^(2/x)]^[lim(x->0)(x²/2)]
=e^0 (应用重要极限lim(x->0)[(1+x)^(1/x)]=e)
=1;
解法二：原式=lim(x->0){e^[xln(1+x/2)]}
=e^{lim(x->0)[xln(1+x/2)]
=e^(0*ln1)
=e^0
=1.