lim[e^x*sinx-x(1+x)]/[x^2*sinx] ,lim[ln(1+x)*ln(1-x)-ln(1-x^

1.lim(x->0) [e^x *sinx - x(1+x)] / [x²sinx]
= lim(x->0) [ e^x sinx - x - x² ] / x³ 等价无穷小代换
= lim(x->0) [ e^x (sinx+cosx) - 1 - 2x ] / (3x²) 洛必达法则
= im(x->0) [ 2 e^x cosx - 2 ] / (6x)
= lim(x->0) e^x (cosx - sinx) / 3
= 1/3
2.lim(x->0) [ln(1+x)*ln(1-x) - ln(1-x^2)] / x^4
= lim(x->0) [ ln(1-x) / (1+x) + ln(1+x) / (x-1) - 1/(x-1) - 1/(x+1) ] / (4x³) 洛必达法则
= lim(x->0) [ (x-1) ln(1-x) + (x+1) ln(1+x) - (x+1) - (x-1) ] / (- 4x³)
= lim(x->0) [ 1 + ln(1-x) + 1 + ln(1+x) - 2 ] / (-12x²) 洛必达法则
= lim(x->0) [ 1/(x-1) + 1/(1+x) ] / (-24x)
= 1/12