设 x = tan t, 则有 arctan x = t,(x->0 <==> tan t -> 0 <==> t-->0 )
要证明:arctan x ~ x ( x->0) <==>等价于证明: t ~ tan t
证明如下:
lim (t-> 0) sin t ~ t
lim (t-> 0) cos t = 1
tan t = sin t / cos t ~ t/1 ==> tan t ~ t ==> arctan x ~ x
设arctanx=t,换元为 t÷tant 即可