x趋近于0,lim[(e^tanx-e^x)/(x^k)]
=lim{e^x*[(e^(tanx-x)-1]/x^k}
=lim [e^(tanx- x)/x^k]
=lim(tanx- x)/x^k
=lim{[sec(x)]^2-1}/(kx^(k-1))
=lim(tanx)^2/(kx^(k-1))
=lim [x^2/(kx^(k-1))]
2=k-1,k = 3
写得比较详细,具体做时可以简写
若熟悉x趋近于0,tanx- x 等价于 (1/3)x^3 (泰勒级数展开）
=lim(tanx- x)/x^k = (1/3)limx^(3-k)=常数 ,k=3