求解几道高数题Lim x趋于0 [e^x-e^(-x)]/(sinx) 的极限Lim x趋于0[x*cot2x] 的极限

这几题可以用洛必达法则来求,即分子分母同时求导后再求极限：
lim(x→0)[e^x-e^(-x)]/(sinx)
=lim(x→0)[e^x+e^(-x)]/(cosx)
=(e^0+e^0)/cos0
=2
lim(x→0)(x*cot2x)
=lim(x→0)(x/tan2x)
=lim(x→0)[1*(cos2x)^2/2]
=1*(cos0)^2/2
=1/2
lim(x→0){x*[e^(1/x) ]- 1}
=lim(x→0)[x/e^(-1/x)-1]
=lim(x→0){1/[e^(-1/x)*1/x^2]-1}
=lim(x→0)[x^2/e^(-1/x)-1]
=-1
lim(x→0)[(1/x)^tanx]
=lim(x→0){e^[tanx*ln(1/x)]}
=lim(x→0){e^tanx/[1/ln(1/x)]}
=lim(x→0){e^{[(cosx)^(-2)]/[x*(-1/x^2)]}}
=lim(x→0){e^[-x/(cosx)^2]}
=e^(0/1^2)
=1
有点长,有问题再问吧……