求lim x→0 (tan2x-sinx)/x以及lim x→0 (cosx-cos3x)/x^2
问题描述：
求lim x→0 (tan2x-sinx)/x以及lim x→0 (cosx-cos3x)/x^2
答
lim x→0 (tan2x-sinx)/x
=lim x→0 (2tanx/(1-tan²x) -sinx) /x
=lim x→0 (2sinx/(cosx·(1-tan²x)) -sinx) /x
=lim x→0 (sinx/x) · lim x→0 (2/(cosx·(1-tan²x)) -1)
=1× (2/(1·(1-0²)) -1)
=1
lim x→0 (cosx-cos3x)/x^2
=lim x→0 (-sinx+3sin3x)/(2x)
=(1/2) (lim x→0 -sinx/x + 3·lim x→0 sin3x/x)
=(1/2) (-1 + 3·lim x→0 3x/x)
=(1/2) (-1 + 3×3)
=4