【30分】几个简单的高数习题,求详解.

1.原式=(1/2)lim(x->0){[(2x)/sin(2x)]*cos(2x)}
=(1/2)lim(x->0)[(2x)/sin(2x)]*lim(x->0)[cos(2x)]
=(1/2)*1*1 (应用重要极限lim(x->0)(sinx/x)=1)
=1/2;
2.原式=lim(x->0){[(1-cos(2x))/x²]*(x/sinx)}
=2lim(x->0)[(sinx/x)²*(x/sinx)]
=2lim(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；
3.原式=lim(x->∞)[sin(1/x)/(1/x)]
=1 (应用重要极限lim(x->0)(sinx/x)=1);
4.原式=lim(x->∞){[1-2/(2x+1)]^[((2x+1)/2)*((2x)/(2x+1))]}
=e^{lim(x->∞)[(2x)/(2x+1)] (应用重要极限lim(x->0)[(1+1/x)^x]=e)
=e^(1/(1+0))
=e.