Question

等价无穷小limx→0+(1+x^2)^k/1-cosx

Answer 1

问：等价无穷小limx→0+(1+x^2)^k/1-cosx

答：设y=(1+x^2)^[1/(1-cosx)]lny=ln{(1+x^2)^[1/(1-cosx)]}=[1/(1-cosx)]*ln(1+x^2)=ln(1+x^2)/(1-cosx)lim(x→0) lny ＝lim(x→0) ln(1+x^2)/(1-cosx)＝lim(x→0) [ln(1+x^2)]'/(1-cosx)'＝lim(x→0) [2x/(1+x^2)]/sinx＝{lim(x→0) [2/(1+x^2)]}*{lim(x→0) x/sinx}＝2*{lim(x→0) 1/(sinx/x)}＝2/{lim(x→0) sinx/x}＝2/1＝2lim(x→0)（1+x²)^[1/(1-cosx)]=lim(x→0) y=lim(x→0) e^(lny)=e^{lim(x→0) lny}