Question

当x趋近于0时，求(tanx-sinx)/x^3的极限 当x趋近于0时，求1/x-1/(e^x-1)的极限

求详细过程

Answer 1

1. lim (tanx-sinx)/x^3
=lim (sinx-sinxcosx)/(x^3*cosx)
=lim(sinx-sinxcosx)/x^3
=lim (cosx-cos²x+sin²x)/(3x²)
=lim (cosx-cos2x)/(3x²)
=lim (-sinx+2sin2x)/(6x)
=lim (-cosx+4cos2x)/6
=(-1+4)/6
=1/2
2. lim [1/x-1/(e^x-1)]
=lim (e^x-x-1)/[x*(e^x-1)]
=lim (e^x-1)/[e^x-1+x*e^x]
=lim (e^x)/(e^x+e^x+x*e^x)
=lim 1/(2+x)
=1/2

Answer 2

用洛必达法则
(tanx-sinx)/x^3=1/4(x-->0)

1/x-1/(e^x-1)=(e^x-1-x)/(x(e^x-1))=1/2(x-->0)