Question

lim根号下(1－cosx平方)/(1-cosx)=

Answer 1

lim(x->0)[√(1-cos(x²))/(1-cosx)]

=lim(x->0)[√(2sin²(x²/2))/(2sin²(x/2))] (应用三角函数倍角公式)

=lim(x->0)[√2sin(x²/2)/(2sin²(x/2))]

=lim(x->0)[√2*(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*1*1 (应用重要极限lim(t->0)(sint/t)=1)

=√2.