【求解几个简单的高数习题,请详解.1.lim0〔sin2x/sin5x]2.limx～0[tan3x/sin2x]3.limx～0[xcot2x]4.limx～0[1-cos2x]/[xsinx]5.limx～无穷大xsin(1/x)】
求解几个简单的高数习题,请详解.
1.lim0〔sin2x/sin5x]
2.limx～0[tan3x/sin2x]
3.limx～0[xcot2x]
4.limx～0[1-cos2x]/[xsinx]
5.limx～无穷大xsin(1/x)
1.原式=lim(x->0){(2/5)[sin(2x)/(2x)][(5x)/sin(5x)]}
=(2/5)*lim(x->0)[sin(2x)/(2x)]*lim(x->0)[(5x)/sin(5x)]
=(2/5)*1*1(应用重要极限lim(x->0)(sinx/x)=1)
=2/5;
2.原式=(3/2)lim(x->0){[sin(3x)/(3x)][(2x)/sin(2x)][1/cos(3x)]}
=(3/2)*lim(x->0)[sin(3x)/(3x)]*lim(x->0)[(2x)/sin(2x)]*lim(x->0)[1/cos(3x)]
=(3/2)*1*1*1(应用重要极限lim(x->0)(sinx/x)=1)
=3/2;
3.原式=lim(x->0)[(x/sinx)*cosx]
=lim(x->0)(x/sinx)*lim(x->0)(cosx)
=1*1(应用重要极限lim(x->0)(sinx/x)=1);
4.原式=lim(x->0){[(1-cos(2x))/x²]*(x/sinx)}
=2*lim(x->0)[(sinx/x)²*(x/sinx)]
=2*lim(x->0)[(sinx/x)²]*lim(x->0)(x/sinx)
=2*1²*1(应用重要极限lim(x->0)(sinx/x)=1)
=2;
5.原式=lim(x->0)[sin(1/x)/(1/x)]
=1(应用重要极限lim(x->0)(sinx/x)=1).
注：以上各题还可以应用罗比达法则.但应用重要极限简洁明了.