lim|x趋向于0+ ln(1+2x^2)/x^2 = 2,

题目

lim|x趋向于0+ ln(1+2x^2)/x^2 = 2,

答案

解法一：原式=lim(x->0)[(4x/(1+2x²))/(2x)] (0/0型极限,应用罗比达法则)
=lim(x->0)[2/(1+2x²)]
=2/(1+0)
=2；
解法二：原式=lim(x->0){ln[(1+2x²)^(x²)]}
=ln{lim(x->0)[(1+2x²)^(x²)]}
=ln{lim(x->0)[(1+2x²)^(1/(2x²))]²}
=ln{lim(x->0)[(1+2x²)^(1/(2x²))]}²
=ln(e²) (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=2lne
=2.