Question

函数的极限

求解，要过程

Answer 1

原式=lim(x→0) e^x[e^(tanx-x)-1]/(xsin^2x)
=lim(x→0) e^x(tanx-x)/(x^3)
=lim(x→0) (tanx-x)/(x^3)
=lim(x→0) (sec^2x-1)/(3x^2)
=lim(x→0) (1-cos^2x)/(3x^2*cos^2x)
=lim(x→0) sin^2x/(3x^2)
=1/3

Answer 2

当x->0时，分子分母都->0，所以可以使用洛必达法则和等价无穷小代换
原式=lim(x->0) e^x*[e^(tanx-x)-1]/xsin^2x

=lim(x->0) e^x*(tanx-x)/x^3
=lim(x->0) e^x * lim(x->0) (tanx-x)/x^3
=1*lim(x->0) (sec^2x-1)/3x^2
=lim(x->0) (2sec^2xtanx)/6x
=lim(x->0) (xsec^2x)/3x
=1/3

Answer 3