Question

求lim(x→0) (√1-cosx^2)/(1-cosx),还有题lim(x→0) (x-xcosx)/(tanx-s

Answer 1

（1）lim(x->0)[√(1-cos(x²))/(1-cosx)]=lim(x->0)[√(2sin²(x²/2))/(2sin²(x/2))] (应用半角公式)

=√2lim(x->0)[(sin(x²/2)/(x²/2))((x/2)/sin(x/2))²]

=√2{lim(x->0)[sin(x²/2)/(x²/2)]}*{lim(x->0)[(x/2)/sin(x/2)]}²

=√2*1*1² (应用重要极限lim(z->0)(sinz/z)=1)

=√2

（2）lim(x->0)[(x-xcosx)/(tanx-sinx)]=lim(x->0)[x(1-cosx)/(sinx/cosx-sinx)]

=lim(x->0)[x(1-cosx)cosx/sinx(1-cosx)]

=lim(x->0)[(x/sinx)cosx]

=[lim(x->0)(x/sinx)]*[lim(x->0)(cosx)]

=1*1 (应用重要极限lim(z->0)(sinz/z)=1)

=1