大一极限题(要大致过程或方法,谢谢）

Question

1.lim[(tanx-sianx)/x^3] x趋近0
2.lim{x^2[1-cos(1/x)]} x趋于无穷
两道题答案都是1/2

Answer 1

1.原式=lim(x->0)[(sec²x-cosx)/(3x²)] (0/0型，应用罗比达法则)
=lim(x->0)[(1/cos²x)(1-cos³x)/(3x²)]
=[lim(x->0)(1/cos²x)]*{lim(x->0)[(1-cos³x)/(3x²)]}
=1*lim(x->0)[(sinxcos²x)/(2x)] (0/0型，应用罗比达法则)
=lim(x->0)[(sinx/x)(cos²x/2)]
=[lim(x->0)(sinx/x)]*[lim(x->0)(cos²x/2)]
=1*(1/2)
=1/2
2.原式=lim(x->∞){[1-cos(1/x)]/(1/x²)}
=(1/2)lim(x->∞){[sin(1/x)/(1/x)]} (0/0型，应用罗比达法则)
=(1/2)*1 (应用特殊极限lim(x->0)(sinx/x)=1)
=1/2