Question

高数求极限问题

如图

Answer 1

(1)
lim(x->0)[ x-ln(1+tanx)] /(2x^2) (0/0 分子分母分别求导）
=lim(x->0)[ 1- (secx)^2/(1+tanx) ] /(4x)
=lim(x->0)[ 1+tanx - (secx)^2 ] /[(4x) (1+tanx) ]
=lim(x->0)[ 1+tanx - (secx)^2 ] /(4x)
=lim(x->0)[ tanx - (tanx)^2 ] /(4x)
=lim(x->0)[ x - x^2 ] /(4x)
=1/4
(2)
可以，但可能你会不明白！
这个来自泰勒展式
tanx ~ x+ (1/3)x^3
ln(1+tanx )
= tanx -(1/2)(tanx)^2 + o((tanx)^3)
= x +(1/3)x^3 - (1/2)[x+ (1/3)x^3]^2 + o(x^3)
= x -(1/2)x^2 + o(x^2)
x-ln(1+tanx) = (1/2)x^2 +o(x^2)
lim(x->0)[ x-ln(1+tanx)] /(2x^2)
=lim(x->0) (1/2)x^2 /(2x^2)
=1/4