利用等价无穷小和罗比达法则.
e^(x-1)-e^(x^2-1)趋于0,tanx与x等价.又因为分子,分母都趋于0,可以用罗比达法则.
如下
lim tan(e^(x-1)-e^(x^2-1))/(arctanx-π/4)=(e^(x-1)-e^(x^2-1))/(arctanx-π/4)=(e^(x-1)-2xe^(x^2-1))(1+x^2)
然后将x=1代入得
lim tan(e^(x-1)-e^(x^2-1))/(arctanx-π/4)=(e^(x-1)-e^(x^2-1))/(arctanx-π/4)=(e^(x-1)-2xe^(x^2-1))(1+x^2)=-2