lim (1+2x)^(1/x)
=e^[lim (1/x)ln(1+2x)]
=e^[lim ln(1+2x)/x]
0/0型极限,用洛必达法则
=e^[lim (2/(1+2x))/1]
=e^[lim 2/(1+2x)]
=e^[lim 2]
=e^2